THOMSON'S    NEW  >;>*ATHFMATI'"\\T.    SFJME  !. 


THE 


COLLEGIATE  ALGEBRA 


ADAPTED     TO 


COLLEGES    AND    UNIVERSITIES. 


BY 

JAMES    B.    THOMSON,    LL.  D., 

AUTHOR  OF  NEW  MATHEMATICAL  SERIES, 

AND 

ELIHIT    T.    QUIMBY,    A.M., 

LATE  PROFESSOR  OF  MATHEMATICS  IN  DARTMOUTH  COLLEGE. 


OF 

NEW    YORK: 

Clark    &    Matnaed,    Publishers 

5  Barclay  Street. 

chicago:  46  madison  street. 

1SS0. 


T 


TinyMSOF^'llTilEiiTICAL   SEKIES. 


I.  A  Graded  Series  of  Arithmetics,  in  three  Boohs,  viz.  .- 

New  Illustrated  Table   Book,   or   Juvenile  Arithmetic.      With  oral 
and  slate  exercises.     (For  beginners.)     128  pp. 

New    Rudiments    of    Arithmetic.      Combining   Mental    with    Written 
Arithmetic.     (For  Intermediate  Classes.)    224  pp. 

New  Practical  Arithmetic.    Adapted  to  a  complete  business  education. 
(For  Grammar  Departments.)    384  pp. 

II.  Independent  Boohs. 

Key  to  New  Practical  Arithmetic.      Containing  many  valuable  sug- 
gestions.    (For  teachers  only.)     1G8  pp. 

New   Mental    Arithmetic.      Containing    the    Simple    and    Compound 
Tables.     (For  Primary  Schools.)     144  pp. 

Complete   Intellectual   Arithmetic.      Specially   adapted  to  Classes  in 
Grammar  Schools  and  Academies.     168  pp. 

III.  Supplementary  Course. 

New  Practical  Algebra.     Adapted  to  Higb   Schools   and  Academies. 
312  pp. 

Key  to  New  Practical  Algebra.     With  full  solutions.     (For  teachers 
only.)    224  pp. 

New   Collegiate    Algebra.     Adapted  to  Colleges  and   Universities.    By 
Thomson  &  Quimby.     346  pp. 

Complete  Higher  Arithmetic.     (In  preparation.) 

***  Each  book  of  the  Series  is  compleU  in  itself. 


Copyright,  1879,  1880,  by  J.  B   Thomson  and  E.  T.  Qrmr.v. 


Electrotyped  by  Smith  &  McDougal,  S2  Beekman  Street,  New  York. 


P  REFA  CE. 


SOON  after  the  publication  of  the  "  New  Practical  Alg 
bra/'  the  author  was  urgently  requested  by  sever; 
mathematical  professors  to  prepare  a  higher  work  on  the  sam 
general  plan,  adapted  to  the  wants  of  Colleges  and  Univers 
ties.  In  compliance  with  these  requests,  the  present  treat  i; 
was  undertaken  and  is  now  presented  to  the  public. 

To  facilitate  its  preparation  he  was  fortunate  in  securing 
the  co-operation  of  Prof.  E.  T.  Quimby,  of  Dartmouth  College, 
a  gentleman  of  more  than  twenty-five  years  experience  in 
teaching  mathematics. 

The  work  presents  a  full  discussion  of  all  the  subjects 
usually  contained  in  the  most  complete  text-books  in  use,  as 
the  Demonstration  of  the  Binomial  Formula,  the  Computation 
of  Logarithms,  Theory  of  Equations,  Sturm's  Theorem,  Inde- 
terminate Coefficients,  Series,  Infinitesimal  Analysis,  Horner's 
Method  of  Approximation,  Loci  of  Equations,  Exponential 
Equations,  etc. 

A  few  subjects,  as  Probabilities,  etc.,  thought  to  be  of  less 
importance  have  been  thrown  into  an  Appendix.  While  the 
most  approved  authors  have  been  freely  consulted  and  their 
methods  carefully  compared,  the  plan  and  execution  of  the 
work  are  the  results  of  long  personal  experience  in  the 
class-room. 

The  arrangement  is  systematic,  each  subject  appearing  in  its 
natural  order,  and  the  dependence  of  the  principles  upon  each 
other  is  shown  by  frequent  references. 


i  v  P  B  B  1-"  ACE. 

The  Examples  are  numerous,  and  have  been  prepared  with 
a  view  both  to  illustrate  the  principles  under  discussion,  and 
to  stimulate  thoughl  on  the  part  of  the  student. 

Great  pain-  have  been  taken  to  make  the  rules  and  defini- 
tions clear  and  concise,  and  the  demonstrations  simple, 
rigorous,  and  logical.  The  subject  has  been  brought  down 
to  the  present  time,  and  the  best  of  the  improved  methods  of 
teaching  the  various  topics  have  been  adopted,  to  the  exclusion 
of  such  as  are  obsolete. 

Originality  of  mailer  is  not  to  be  expected  in  a  book  of  this 
kind  :  but  it  will  be  found  that  several  subjects  have  been 
treated  in  a  manner  more  or  less  original.  The  reader  is 
referred  t<>  the  Articles  on  the  use  of  the  Directive  Signs,  or 
Factors  of  Direction,  the  treatment  of  Imaginary  Quantities, 
Logarithms,  Scries,  etc. 

The  work  is  designed  to  meet  a  want  which  has  been  felt 
to  a  greater  or  less  extent,  but  which  heretofore  has  not  been 
supplied  in  a  satisfactory  manner. 

It  is  earnestly  commended  to  the  attention  of  instructors 
ami  students,  with  the  hope  that  it  will  be  found  to  contain 
enough  that  is  new  and  useful  to  satisfy,  in  a  measure,  the 
views  of  those  who  believe  in  progress,  and  who  have  desired 
some  departures  from  the  beaten  track. 

In  conclusion,  the  authors  would  avail  themselves  of  the 
opportunity  to  express  their  obligations  to  their  friends  who 
have  favored  them  with  many  valuable  suggestions  upon  the 
subji 

Brooklyn,  X.  Y.,  ./    y,  1S79. 


CONTENTS. 


PAGE 


Introduction, 9 

Definitions, 9-15 

Functions  of  Quantities, 10 

Axioms,       .                 15 

Notation,       .                       16 

Symbols  of  Quantity, 16 

Symbols  of  Operation, 17 

The  Distinction  between  Factors  and  Terms,     .        .  18 

Symbols  of  Eelation, 20 

Symbols  of  Abbreviation,    ......  20 

Algebraic  Expressions, 21 

Positive  and  Negative  Quantities,      .        .        .        .  22 

Directive  Signs, 23 

The  Directive  Force  of  the  Signs  +  and  — ,     .        .  25 

Rule  for  the  Directive  Signs, 26 

Exercises  in  Notation, 27 

Algebraic  Addition, 29 

General  Eule, 30 

Adding  Similar  Polynomials, 32 

Algebraic  Subtraction, 32 

General  Rule, t>2> 

Use  of  Parentheses, 34 

Multiplication, 36 

Multiplication  of  Monomials, 37 

Signs  in  Multiplication,          .                          ...  38 

Multiplication  of  Polynomials, 39 

Multiplication-  by  Detached  Coefficients,        .        .        .  40 

Division,  .........  41 

Dividing  a  Monomial  by  a  Monomial,    .        .        .        .  42 


vi  CONTENTS. 


PACK 


Analogy  between  Coefficient  and  Exponent,      .        .  43 

Signa  in  Division, 45 

Dividing  a  Polynomial  by  a  Monomial,      ...  46 

Dividing  a  Polynomial  by  a  Polynomial,        .        .        -  47 

Dividing  by  Detached  Coefficients,     ....  50 

S3  nthetic  Division.          .                 51 

Factoring,  -  53 
Theorems  for  Factoring  Binomials,  .  .  .  .53 
Finding   the   Binomial   or   Polynomial   Factors   of    a 

Polynomial,. 55 

Greatest  Common  Divisor, 58 

Least  Common  Multiple, 6r 

Fractions, 63 

Signs  of  Fractions, 64 

Reduction  of  Fractions, 65 

Addition  and  Subtraction  of  Fractions,     ...  69 

Multiplication  of  Fractions, 70 

Division  of  Fractions, 72 

General  Rule, 74 

A  Finite  Quantity  divided  by  Zero,  as  a  -4-  o,    .        .  75 

An  Infinitesimal  divided  by  a  Finite  Quantity,  as  o-j-a,  75 

Equations  of  the  First  Degree,     ...  77 

Conditional  Equations, .        .  78 

Reduction  *4'  Equations, 7S 

Methods  of  Reducing  Equations  of  the  First  Degree,    .  79 

Proof  of  Reduction  of  Equations,      ....  81 

Solution  of  Problems, 83 

Discussion  of  Problems, 85 

Problem  of  the  Couriers, 86 

Sim  ultaneous  Equations  of  the  Fii'st  Degree,  91 

Elimination, 92 

Three  Unknown  Quantities, 98 

Powers  and  Hoots, 101 

Radical  and  Rational  Quantities  defined,   .        .        .  102 

\  Surd  or  Irrational  Quantity 102 

Powers  of  Monomials. 103 

Towers  of  Binomials, Ic>4 


CONTENTS.  vii 


PAGE 


Binomial  Formula, 104 

Involution  of  Polynomials,     .        .        .        .  -  io5 

Multinomial  Theorem, 105 

Evolution  of  Polynomials, 107 

Signs  of  Boots, 109 

Imaginary  Quantities, 113 

Calculus  of  Radicals, 120 

Multiplication  of  Radicals, 121 

Division  of  Radicals,   .        .        .        .        .        .        .122 

Former  Notation  of  Imaginary  Quantities,     .        .        .  1 24 

Reduction  of  Radicals, 126 

Radical  Equations, 131 

Equations  of  the  Second  Degree, .       .       .      132 

A  Complete  Equation, 133 

Completing  the  Square, 134 

Higher  Equations  solved  by  Quadratics,        .        .        .140 
Problem  of  the  Lights,        .        .        .        .        .        .143 

Simultaneous  Equations  of  the  Second  Degree,     .        -  '45 
Inequations,    .       .  .       .       .       .151 

Reduction  of  Inequations, 152 

Ratio  and  Proportion, 154 

Theorems, 155-160 

Permutations  and  Combinations,       .  163 

Formulas, 164 

Infinitesimal  Analysis, 166 

Notation,         . .168 

Differential  Coefficient, .169 

Differentiation,       ....  ...  170 

Indeterminate  Coefficients,      .  .       .176 

Development  of  Functions, 177 

Decomposition  of  Fractions,        .  180 

Demonstration  of  the  Binomial  TJieorem,    .  184 

Logarithms,     .        .  .        .  .187 

Briggs'  System,       .        .        .        .        .        .        .        .  191 

Table  of  Logarithms, 193 

To  Multiply  and  Divide  by  Logarithms,         .        .    195,  196 

Computation  of  Logarithms,   ....      200 


CONTENTS. 


FACJ 


Scries,         .                207 

Interpolation  <>!  T<nn~. 208 

Equidifferenl  Serii  -. 210 

Sum  of  the  Terms,         .        - 211 

Formulas  of  Eq indifferent  Series,      ....  216 

Recurring  Series,    ...--...  218 

Equimultiple  Scries,  ....                 .  222 

Harmonic  Series, 226 

Development  of  Formulas, 227 

I;      rsion  of  Series, 231 

Loci  of  Equations, 232 

I  definitions -  2^^ 

Theory  of  Equations,        .....  239 

Numerical  Higher  Equations, 240 

Divisibility, 243 

Number  of  Roots,  .                245 

Formation  of  Equations, 245 

Forms  <>{'  Roots,      .                  247 

Signs  of  Root-. .250 

Limits  of  Roots, 252 

Limiting  Equation,    .        .        .        .                 .        .  254 

Equal  Roots, 255 

Commensurable  Roots 258 

Incommensurable  Roots, 263 

Sturm's  Theorem, 264 

Horner's  Method  of  Approximation 267 

Recurring  Equations, 27.) 

Binomial  Equations 277 

Exponential  Equations, 279 

Appendix, 280 

Probabilities 280 

( !ardan's  Formula, 282 

I I  jcartes'  Formula,  286 

ContiDUi  '1  Fractions .  288 

M    cellaneous  Problems, 298 

Formulas, 303 

No  305 


ALGEBRA. 


INTKODUCTIOlSr. 

Art.  1.    Mathematics  is  the  science  of  quantity. 

2.  Quantity  is  anything  which  can  be  measured;  as, 
distance,  space,  time,  etc. 

3.  The  Measure  of  a  Quantity  is  the  number  of 
times  it  contains  another  quantity  of  the  same  kind,  called  the 
unit  of  measure ;  or,  it  is  the  ratio  of  the  quantity  to  the  unit 
of  measure.     Hence, 

4.  Number  is  the  measure  of  quantity. 

5.  A  quantity  is  measured  mechanically  by  applying  the 
unit  of  measure  directly  to  the  quantity,  and  counting  the 
number  of  times  it  is  applied. 

Thus,  the  application  of  the  yard-stick  to  measure  the  length  of  a 
piece  of  cloth,  and  of  the  surveyor's  chain  to  measure  distances,  are 
examples  of  the  mechanical  measurement  of  quantities. 

6.  When  the  unit  is  not  contained  in  the  quantity  to  be 
measured  an  integral  number  of  times,  this  unit  may  be 
divided  into  any  number  of  equal  parts,  and  one  of  these  parts 
taken  as  a  unit.  In  this  way,  a  fraction  or  a  mixed  number 
may  express  the  measure  of  a  quantity. 

Thus,  we  may  have  a  piece  of  cloth  5^  or  5  J  yards  long1.  The  exact 
measure  of  a  quantity  cannot  be  found  with  a  unit  which  cannot  be 
divided  into  such  a  number  of  equal  parts  that  one  of  these  parts  shall 
be  contained  an  integral  number  of  times  in  the  quantity  to  be  measured. 

7.  Commensurable  Quantities  are  those  that  cnn 
be  measured  mth  the  same  unit. 


1°     .*.  :  :  .■  •••  .....XN^b.od-ijction. 

8.  Incommensurable' Quantities  are  those  that 
cannot  be  measured  with  the  same  unit. 

Thus,  the  sick  of  a  square  and  its  diagonal  are  distances  that  mnnotf 
fa  /„,,/.*///•,</  tottf  thi  same  unit.  Bowevershort  the  unit  may  be  with 
which  we  attempl  to  measure  these  two  lines,  if  it  give  an  exact  measure 
of  one,  it  will  not  < >l"  the  other. 

9.  A  Single  Quantity  is  called  commensurable  or 
incommensurable  according  as  it  can  or  cannot  he  measured 
with  the  unit  we  are  using. 

10.  Quantities  which  are  of  different  hinds  or  natures, 
as  time  and  distance,  cannot  be  compared  one  with  the  other; 
for,  the  magnitude  or  extent  of  one  is  wholly  unlike  that  of  the 
other,  so  thai  one  cannot  he  made  the  unit  of  measure  for  the 
other. 

11.  Any  quantity  may  he  made  the  unit  of  meas- 
ure for  quantities  of  its  own  hind,  but  for  the  purposes  of 
truth  and  other  mutual  uses,  the  unit  must  he  generally  known 
ami  accepted  ;  hence  the  necessity  of  units  established  by  law 
as  a  national  standard. 

Note. — In  common  language,  we  speak  of  measuring  apile  of  wood, 
or  a  piece  of  lifnd,  but,  strictly  speaking,  that  which  we  reuUy  meas- 
ure is  the  space  occupied  by  the  wood,  and  the  area  of  the  land.  So 
also  we  measurt  the  It  ngth,  the  /reight,  the  density,  the  elasticity,  etc.,  of 
mati  rial  bodies,  but  not  the  bodies  themselves. 

12.  Quantities  as  they  appear  in  nature  have  certain  defi- 
nite nla! ions  to  each  other,  which  make  them  mutually 
dependent,  and  from  which  the  measure  of  one  may  be  found 
when  the  measures  of  others  arc  known. 

Mathematics  investigates  these  relations  and  determines  the  measures 
of  quantities  indirectly,  or  without  direct  measurement. 

13.  Quantities  thus  mutually  dependent  arc  called 
Functions  of  each  other.     For  example, 

14.  In  all  motion,  three  (piantities  are  involved,  viz.: 
Time.  Distance,  and  Velocity,  These  (piantities  are 
-<>  related  that  neither  can  change  without  changing  one  or 
both  of  the  others.  If  the  velocity  increase,  the  distance  will 
increase  or  the  time  decrease,  or  both  these  results  may 
follow. 


INTRODUCTION.  11 

Notes. — i.  In  common  language,  we  say  the  time  depends  on  the 
distance  and  velocity ;  the  distance  on  the  time  and  velocity;  and  the 
velocity  on  the  time  and  distance. 

2.  In  mathematical  language,  the  time  is  a  function  of  the  distance. 
and  velocity  ;  the  distance  is  a  function  of  the  time  and  velocity ;  and 
the  velocity  is  &  function,  of  the  ft'm«  and  distance. 

15.  The  question,  ''What  function  is  one  quantity  oi 
others  ?  "  refers  to  the  manner  in  which  the  latter  quantities 
must  be  combined  or  treated  to  give  the  measure  of  the 
former;  as,  "What  function  is  the  distance  of  the  time  and 
velocity?"  The  answer  to  which  would  be,  "  TJte  product"; 
that  is,  the  distance  is  the  product  of  the  time  by  the  velocity. 

16.  In  like  manner,  let  the  pupil  answer  the  following 
questions : 

r.  What  function  of  the  side  of  a  square  is  its  area  ? 

2.  What  function  of  the  radius  of  a  circle  is  its  diameter? 

3.  What  function  of  the  diameter  of  a  circle  is  its  circum- 
ference ?    Its  area  ? 

4.  What  function  of  the  principal,  rate  per  cent,  and  time, 
is  the  interest  on  a  note  ?     The  amount  ? 

5.  What  function  of  the  itumber  of  pounds  and  the  price 
per  pound  is  the  cost  of  an  article  ? 

6.  What  function  of  its  sides  is  the  area  of  a  rectangle  ? 

17.  A  Pvojwsit  ion  is  something  proposed  for  demon- 
stration, or  solution. 

18.  A  TJieovem  is  a  proposition  /or  demonstration. 

19.  A  Problem  is  a  proposition  for  solution. 

Note. — A  theorem  affirms,  "  This  is  true,"  and  requires  demon- 
stration. 

A  problem  inquires,  "  What  is  true  ?  "  and  requires  solution. 

20.  An  A.riom  is  a  self-evident  theorem. 

21.  A  Postulate  is  a  self-evident  problem.. 

Note. — A  truth  is  called  self-evident  when  it  commands  the  instant 
assent  of  one  who  is  acquainted  with  the  subject  to  which  it  relates,  and 
cannot  be  made  plainer  by  any  proof, 


12  INTRODUCTION. 

22.  A  Demonstration  is  an  arrangement  of  defi  nitions, 

((.minis,  and  postulates,  by   which  the  truth  of  a  theorem  is 
established. 

Note.— A  direct  demonstration  proves  that  a  theorem  is  t vue,  by 
assuming  the  truth  of  certain  definitions  and  axioms,  and  from  these 
premises  deducing  other  truths,  till  we  arrive  at  the  one  which  is  to  be 
established. 

An  indirect  demonstration  proves  that  a  theorem  is  not  untrue,  by 
proving  that  the  supposition  of  its  contrary  involves  an  absurdity. 

23.  A  Solution  is  an  arrangement  of  axioms  and  postu- 
lates by  which  the  answer  to  &  problem  is  determined. 

24.  The  Hypothesis  of  a  proposition  is: 

ist.  Iii  a  theorem; — The  conditions  on  which  the  theorem 
is  affirmed. 

2d.  In  a  problem ; — TJie  data  from  which  the  required 
truth  is  to  be  determined. 

25.  A  Corollary  is  an  inference  from  a  preceding 
demonstration  or  solution. 

26.  An  Equation  is  aw  expression  of  equality  between 
two  quantities. 

The  equation  is  used  to  express  in  algebraic  language  the 
relations  between  quantities  which  are  functions  of  each  other. 

27.  Known  Quantities  are  those  from  which  other 
quantities  are  to  be  determined.  Arbitrary  values  may  there- 
to it  be  assigned  to  them  at  pleasure. 

28.  Unknown  Quantities  are  those  whose  values 
are  to  be  determined  from  their  relations  to  other  quantities. 
They  are  regarded  d&  functions  oi  known  quantities  and  cannot 
therefore  have  arbitrary  values  assigned  to  them. 

29.  Problems  are  of  two  kinds: 

[st  Those  which  require  some  geometrical  or  mechanical 
Iruction  :    as,  To  construct  a  triangle  from   three  given 


INTRODUCTION.  13 

2d.  Those  which  require  the  measure  of  a  quantity  from 
its  relations  to  other  quantities;  as,  To  hud  the  base  of  a 
right-angled  triangle  from  the  other  sides. 

30.  The  Solution  of  a  problem  of  the  second  kind  is 
made  up  of  three  distinct  parts  or  steps: 

1st.  Finding  the  equations  which  express  the  relations 
between  the  quantities  involved. 

2d.  Finding  from  these  equations  ivliat  function  the 
unknown  quantity  is  of  the  known  quantities. 

3d.  Substituting  in  this  function  the  numbers  representing 
the  known  quantities. 

31.  The  first  of  these  steps  requires  a  knowledge  of 
that  branch  of  mathematics  or  physics  to  which  the  problem 
belongs;  as,  Geometry,  Mechanics,  etc. 

The  second  is  the  province  of  Algebra. 
The  third  belongs  exclusively  to  Arithmetic. 

32.  For  illustration  take  the  following  problem : 

A  rope  50  feet  long  attached  to  the  top  of  a  vertical  pole, 
reaches  the  ground  40  feet  from  the  foot  of  the  pole,  on  a 
horizontal  plane;  how  high  is  the  pole? 

First  step  :  By  geometry  we  learn  that  the  square  of 
the  length  of  the  rope  equals  the  sum  of  the  squares  of  the 
length  of  the  pole  and  the  distance  on  the  ground.  This  rela- 
tion expressed  by  an  equation  is 

a?  =  b°~  +  z2, 

in  which  the  letters  a,  b,  and  x  have  been  put  for  the  length 
of  the  rope,  distance  on  the  ground,  and  height  of  the  pole, 
respectively. 

Second  step  .'  By  algebra  this  equation  is  reduced  to  the 
form 

x  =  's/a2  —  b~, 

which  shows  how  the  known  quantities  must  be  combined  to 
produce  the  unknown;  in  other  words,  what  function  the 
unknown  is,  of  the  knoini. 


1-4  I  N  I  BO  I)  UC  Tlo.N. 

Third  Step :    By  arithmetic  the  numbers  50  and  40  are 

substituted  for  a  ami  b  :  thus, 


x  =  Vsoi  —  4°2  —  3°>  Ans. 

33.  The  equation  a'2  =  b2  +  x2  cannot  be  true  unless  the 
quantities  a,  b,  and  x  are  mutually  dependent;  thai  is,  unless 
eac//  is  a  function  of  the  o^fer  £k>o.  The  equation  therefore 
Implies  that  a;  is  a  function  of  a  and  £,  but  does  not  state 
explicitly  what  function. 

In  such  an  equation  a;  is  said  to  be  an  Implicit 
Function  of  a  and  J. 

The  equation  , 

1  x  =  Va>  -  V2 

states  explicitly  what  function  x  is  of  a  and  b,  and  it  is 
therefore  called  an  Explicit  Function. 

34.  This  change  from  an  implicit  to  an  explicit  function 
is  called  "  reducing  the  equation." 

The  explicit  function  resulting  from  the  reduction  of  the 
equation  is  called  a  Formula,  and  is  the  expression  in 
algebraic  language  of  an  arithmetical  rule. 

The  above  formula  for  finding  the  perpendicular  of  a 
right-angled  triangle  when  the  hypothenuse  and  base  are 
given,  being  translated  into  common  language,  becomes  the 

I'm  1  In  wing. 

I!i  1.1;. — Subtract  the  square  of  the  base  from  the  square  of 
the  hypothenuse,  and  take  the  square  root  of  the  difference. 

From  the  preceding  illustrations  Ave  have  the  following 
ilelinii  ions  : 

35.  Algebra  is  the  science  of  the  Equation.     (Art.  34.) 
[ts  objeel  is  the  reduction  of  equations,  by  which  formulas 

for  arithmetical  computations  are  obtained. 

36.  Arithmetic  is  the  Science  of  Numbers.  Its  objeel 
is  the  substitution  of  numbers  in  algebraic  formulas;  or,  what 
is  equivalenl  to  this,  the  combination  of  numbers  in  accordance 
with  rules  furnished  by  Algebra, 


INTRODUCTION.  15 

37.  The  Reduction  of  an  Equation  consists  in  such 
transformations  as  will  make  the  unknown  quantity  an  explicit 
function  of  the  known  quantities. 

38.  The  Reduction  of  Equations  is  based  on  the 
following 

AXIOMS. 

i°.    Equal  quantities  equally  affected  remain  equal. 

2°.    Equal  quantities  unequally  affected  become  unequal. 

3°.    Unequal  quantities  equally  affected  remain  unequal. 

4°.  Quantities  equal  to  the  same  quantity  are  equal  to  each 
other. 

5°.  Quantities  differing  equally,  in  both  magnitude  and 
direction,  from  the  same  quantity  are  equal  to  each 
other. 

6°.  TJie  whole  is  greater  than  its  part,  and  is  equal  to  the 
sum  of  all  its  parts. 


CHAPTER     I . 

NOTATION. 

39.  Notation  in  Algebra  is  the  method  of  expressing 
quantities,  their  relations,  and  combinations,  by  general  sym- 
bols. The  algebraic  symbols  differ  from  the  Arabic  figures,  or 
numerical  measures,  in  this  respect;  the  latter  represent 
specific  quantities,  the  former  general  quantities. 

Note. — This  notation  constitutes  what  is  called  the  algebraic  language. 
It  is  necessarily  general,  since  its  object  is  to  furnish  general  formulas  for 
arlthm,  Ural  compvtations. 

40.  The  Symbols  used  may  be  classified  as  follows: 
ist,  Symbols  of  Quantity  ;  2d,  of  Operation;  3d,  of  Relation; 
4th,  of  Abbreviation. 

SYMBOLS      OF      QUANTITY. 

41.  Quantities  ore  commonly  represented  by 

the  letters  of  the  alphabet  Any  letter  may  be  used  to  repre- 
-.  ni  any  quantity,  and  the  same  letter  may  represent  different 
quantities,  subject  to  one  limitation;  the  same  letter  must 
always  stand  for  the  same  quantity  throughout  the  same 
discussion. 

42.  For  uniformity  and  convenience,  the  following  order 
should  be  observed : 

1st.  It  is  customary  to  employ  the  "first  loiters  of  the  alpha- 
be!  to  represenl  known  quantities,  as,  a,  b,  c,  etc.;  and  the 
last  for  unknown  quantities,  as,  x,  y,  :.  etc. 

2d.  Initial  letters  are  frequently  used:  as,  r  or  7?  for 
nut  ins ;  c  for  circumference  j  s  for  sum,  etc. 


NOT  ATION.  17 

3d.  Different  Quantities  of  the  same  kind  may  be 
represented,  in  the  same  problem,  by  the  same  letter,  with 
accents  or  subscript  figures  to  distinguish  the  different 
quantities. 

Thus,  when  a  problem  involves  the  radii  of  several  circles,  we  may 
use  ?■',  r",  /"'",  etc.  (read,  "r  prime,"  "r  second,"  "r  third,"  etc.),  or  rlf 
r.,,  r&,  etc.  (read,  "  /•  sab  one,"  "  ;•  sub  two,"  etc.). 

4th.  The  Greek  letters  are  commonly  used  to  represent 
angles,  but  sometimes  other  quantities. 

Thus,  the  Greek  .i  is  used  for  the  ratio  of  the  circumference  of  a 
circle  to  its  diameter. 

5  th.  The  symbol  a>  (the  figure  8  placed  horizontally) 
represents  infinity,  or  a  quantity  greater  than  any  assign- 
able quantity. 

6th.  The  symbol  o  (zero)  represents  an  infinitesimal  quan- 
tity, or  a  quantity  less  than  any  assignable  quantity. 

43.  Quantities  when  expressed  by  numbers  are  called 
numerical;  when  expressed  by  letters,  they  are  called 
literal  quantities, 

SYMBOLS      OF     OPERATION. 

44.  The  Fundamental  Operations  in  Algebra  are 
Addition,  Subtraction,  Multiplication,  Di- 
vision,  Involution,   and   Evolution. 

45.  Addition  and  Subtraction  are  expressed  as  in 
Arithmetic  by  the  signs  +  and  —  ;  as,  a  +  b  —  c,  read,  "  a 
plus  b  minus  c." 

Quantities  connected  by  the  signs  +  and  —  are  called 
Terms. 

46.  When  the  same  term  is  to  be  added  or  subtracted  more 
than  once,  a  number  is  placed  before  it  to  show  how  many 
times  it  is  to  be  used. 

Thus,  a  +  b  +  b— c— c— c  may  be  written  a  4- 26— 3c. 
A  number  thus  used  to  show  how  many  times  a  quantity 
is  taken  as  a  term  is  called  a  Coefficient. 


IS  N  O  T  A  T  I  0  X  . 

47.  A  Coefficient  may  be  integral  or  fractional,  the 
latter  showing  what  />"/•/  of  a  quantity  is  taken  as  a  term; 
us  3a    I«/'.      It  '":IV  a'6u  'je  numerical,  or  literal,  or  both. 

Thus,  in  the  expression  211b,  2a  may  be  regarded  as  indicating  how 
ma  \y  times  b  is  taken  :  or  2  how  many  times  ab  is  taken,  or  b  how  many 
times  2a  is  taken. 

Note. — The  word  coefficient,  however,  usually  refers  to  the  numeri- 
cal factor  of  a  term.     Hence,  generally, 

48.  A  Coefficient  is  the  factor  or- factors  of  a  term  indi- 
cating the  number  of  times  the  rest  of  the  term  is  taken,  or, 
what  equal  terms  are  taken. 

When  no  coefficient  is  expressed,  1  is  always  understood. 

49.  The  double  sign  ±  is  used  when  a  quantity  may  be 
either  added  or  subtracted,  and  is  read,  "plus  or  minus." 

Thns,  a  ±  b  means  that  the  conditions  of  the  problem  will  be  satisfied 
either  by  adding  or  subtracting  b. 

50.  A  Posit  ire  Quantity  is  one  whose  sign  is  +. 

51.  A  Negative  Quantity  is  one  whose  sign  is  — . 

Note. — The  signs  +  and  —  in  Algebra  have  a  more  general  mean- 
ing than  merely  addition  and  subtraction,  which  will  be  explained  in  the 
proper  place  (Arts.  81-94). 

MULTIPLICATION    AND     DIVISION. 

52.  Quantities  used  as  multipliers  or  divisors  are  called 
Wactors,  in  distinction  from  Terms,  which  are  added  or 
subtracted. 

53.  Multiplication  is  expressed: 

1st.  By  the  usual  sign,  x  ;  as,  2  xaxbxc. 
2d.   By  the  period ;  as,  2-4-6  =  2x4x6. 
3d.  By  writing  the  factors  one  after  the  other  without  any 
sign  :  as,  2a.be  =  2  x  a  x  b  x  c. 

\.'ii.— It  is  customary  to  write  numerical  factors  first,  and  literal 
(acton  after,  in  alphabetical  order,  as,  ytbex. 

54.  Division  \-  expressed  : 

ist.   By  the  usual  sign,  -:-  :  as.  a  -t- b. 


NOTATION.  19 

2d.  By  writing  the  divisor  under  the  dividend  in  the  form 
of  a  fraction ;  as,  -r  =  a-i-b. 

3d.  By  a  colon    :  ;    as,  a  :  b  =  a  -j-  b. 
4th.  By  a  negative  exponent;  as,  ab~x  =  a-i-b,  as  seen 
below. 

55.  When  the  same  factor  is  used  more  than  once,  either 
as  a  multiplier  or  divisor,  a  figure  called  an  Exponent  is 
placed  a  little  above  and  to  the  right  of  it,  to  show  how  many 

times  it  is  used. 

aaci  cfi 

Thus,  instead  of  aaabb  we  write  «362,  and  for  ---  we  write  r- ,  or 

bb  ¥ 

a36-2  (Art.  54,  4th). 


INVOLUTION     AND     EVOLUTION. 

56.  Involution  is  the  multiplication  of  equal  factors. 

57.  Evolution  is  the  process  by  which  a  cpiantity  is 
separated  into  equal  factors. 

58.  A  Power  is  the  product  of  any  number  of  the  ecpial 
factors  of  a  quantity,  and  is  expressed  by  an  exponent ;  as,  a% 
a3,  a*,  etc.,  read,  "  a  second  power "  or  "  a  square,"  "  a  third 
power"  or  "a  cube,"  "a  two-thirds  power,"  etc. 

Note. — In  reading  poioers,  do  not  omit  tlie  word  "power,"  reading 
"  a  fourth,"  "  a  third,"  etc.,  for  that  means  a"" ,  a'" ,  etc. 

59.  A  Root  is  one  of  the  equal  factors  of  a  quantity. 

It  is  therefore  a  power  whose  exponent  is  a  fraction  with  1 
for  a  numerator  ;  as,  a?,  a?,  which  may  be  read,  "  the  square 
root  of  «,"  "  the  cube  root  of  a,"  or  "  a  one-half  power," 
"a  one-third  power." 

60.  The  Denominator  of  a  fractional  exponent  shows 
the  number  of  equal  factors  into  which  the  quantity  is  sepa- 
rated. 

61.  The  Numerator  shows  lioiv  many  of  those  factors 
are  taken.     Hence, 

62.  An  Exponent  shows  what  equal  factors  are  taken. 


20  NOTATION. 

63.  The  Radical  Sign,  V  ,  is  often  used  to  express 
.  a  figure  being  written  over  it  to  indicate  what  root  is 

taken. 

Thus,    y/a  =  a*;     tya  =  a*  ;     y'a'-'  =  a*.      When  no  figure  is 

written  over  the  sign,  2  is  understood  ;  as,  -y/a  =  y/a  =  a?. 

Notk. — The  figure  placed  over  the  radical  sign  is  called  the  Index 
of  the  root,  because  it  denotes  the  name  of  the  root. 

SYMBOLS     OF     RELATION. 

64.  The  Sign  of  Equality  is  =  ;  as,  a  =  t  +  c, 
read,  "a  equals  b  +  c." 

65.  Inequality  is  expressed  by  two  lines  forming  an 
acute  angle,  and  opening  towards  the  greater  quantity;  as, 
a  >  b,  or  a  <  b,  read,  "a  is  greater  than  b"  or  "a  is  less 
than  b." 

66.  The  Sign  of  Variation  is  oc .  It  shows  that  the 
quantities  between  which  it  is  placed  have  a  constant  rat  in. 

Thus,  x  x  //,  read,  "  x  varies  as  y,"  means  that  however  x  may  change 
its  value,  y  also  changes,  so  that  the  quotient  x-i-y  does  not  cliange. 

SYMBOLS     OF     ABBREVIATION. 

67.  The  symbol  .*.  is  used  for  the  word  therefore,  and  V 
for  becan 

68.  The  Vinculum,  horizontal,  ~  ,  or  vertical, 
the  Parenthesis,  (  ),  Brackets,  [  ],  and  Braces,  { 

are  used  to  connect  several  quantities  with  the  same  coefficient, 
exponent,  or  sign. 

'I'h  us,      \[(a      b  —  c)°  —  x  +  y  +  z~\   —  (x  —  a)\  -,     indicates 

ist.  Thai  the  quantity  a  +  b— c  is  to  be  squared. 

2d.  That  the  quantity  x  \-y+s  is  to  be  squared. 

3d.  That  the  second  of  these  squares  is  to  be  subtracted  from  the 
first,  and  the  difference  raised  to  the  third  power. 

.jtli.  Thai  the  quantity  (a*— a)  is  to  be  subtracted  from  this  cube,  and 
(he  square  rool  of  the  difference  taken. 


DOTATION".  21 

It  is  sometimes  convenient  to  use  the  vinculum  vertically  ;  as,     ax 

which  is  the  same  as  (a  +  b— c)x.  +  ° 

— c 

69.  In  writing  a  series  of  terms,  or  factors,  the  expression 
is  often  abbreviated  by  omitting  a  part  of  the  quantities,  where 
they  can  be  easily  supplied,  and  indicating  the  omission  by  a 
succession  of  dots  or  short  dashes. 

Thus,  i  +  2  4-  3  ....  8  ;  meaning  the  sum  of  the  numbers  i,  2,  3,  to  8, 
inclusive. 

The  product  of  the  numbers  1,  2,  3,  etc.,  to  any  given  number  may- 
be expressed  in  this  manner  (1.2-3....  10),  but  it  is  usually  still  further 
abbreviated   by  writing   the   last   factor ;    thus, 

|io  —i  -2-  3  ....  10. 

This  is  read,  "factorial  10,"  meaning  the  product  of  the  natural  numbers 
from  1  to  10  inclusive.     So 

\n  (read  "  factorial  n")  =  1 . 2  •  3 . . . .  n. 


ALGEBRAIC     EXPRESSIONS. 

70.  An  Algebraic  Expression  is  any  quantity  ex- 
pressed in  algebraic  language;  as,  3a,  5a  —  7  b,  etc. 

71.  Tbe  Terms  of  an  algebraic  expression  are  the  quan- 
tities which  are  connected  by  the  signs  +  and  — . 

Thus,  in  a  +  b  there  are  two  terms;  in  x  +  y  xs  —  a  there  are 
three,  y  x  2  being  a  single  term.  For,  quantities  connected  by  the  signs 
x  or  -5-  do  not  constitute  separate  terms. 

72.  A  Monomial  is  an  algebraic  expression  containing 
only  one  term ;  as,  a,  tab,  etc. 

73.  A  Binomial  has  two  terms ;  as,  a  +  I. 

74.  A  Trinomial  has  three  terms ;  as,  a  +  b  +  c. 

75.  A  Polynomial  has  tkvee  or  more  terms  ;  as, 

ax  +  by  —  z  —  x. 
Note. — A  binomial  is  -wwaetrfnes-  carted  a  polynomial. 


22  NOTATION. 

76.  The  "Degree  of  a  term  is  the  number  of  its  literal 
factors.  As  the  number  of  these  factors  is  indicated  by  the 
exponents,  the  degree  of  a  term  will  be  the  sum  of  the  exponents 
of  Us  literal  factors. 

Thus,  2til>,  211-,  and  3«.r  are  of  the  second  degree,  and  a-b,  ah1*,  and  ¥ 
are  of  the  third  degree. 

77.  A  Homogeneous  Polynomial  has  all  its  terms 
of  the  same  degree. 

Thus,  ab-  +  rt26  +  b3  is  homogeneous,  but  a2&-  —  2ab  +  b-  is  not  homo- 
geneous. 

78.  Like  or  Similar  Terms  are  those  containing  the 
same  powers  of  the  same  letters  ;  as,  d~x  and  zdrx. 

79.  Unlike  or  Dissimilar  Terms  contain  different 
Utters,  or  the  same  letters  with  different  exponents. 

Thus,  2a-x  and  2oa?,  2d-  and  2a-x,  a'2x-  and  «-.*%,  etc.,  are  dissimilar 
terms. 

80.  The  Reciprocal  of  a  quantity  is  unity  divided  by 
tli  at  quantity. 

Thus,  a  and  -  are  reciprocals  of  each  other. 


POSITIVE    AND     NEGATIVE    QUANTITIES. 

81.  The  signs  +  and  —  are  used  to  indicate  addition 
and  subtraction  ;  but  if  we  inquire  why  certain  quantities  in 
the  solution  of  a  problem  are  added  and  others  subtracted,  we 

shall  find  that  it  depends  on  the  direction  of  the  quantities. 
This  will  he  best  illustrated  by  a  few  examples. 

ist.  A  man  walks  several  distances,  some  north  and  some 
south.     II"\\  far  north  of  his  starting-point  does  he  stop? 

To  answer  this,  we  add  n<>rtlt  distances  and  subtract  south,  because 
the  former  increast ,  while  the  latter  decrease  the  result, 

I  low  far  south  of  his  starting-point  does  he  stop? 

Here  we  add  south  distances  and  subtract  north,  for  the  same  reason 


NOTATION.  23 

2d.  A  man  has  bills  payable  and  bills  receivable.  What  is 
their  net  value  to  him  ? 

Add  bills  receivable  and  subtract  bills  payable.  They  have  opposite 
directions;  one  represents  cash  coming  to  him  and  the  other  going  from 
him. 

What  is  the  net  amount  of  debt  these  bills  represent  ? 

Add  bills  payable  and  subtract  bills  receivable,  the  question  having- 
been  reversed. 

3d.  What  is  the  value  of  my  bank  account  ? 

Add  deposits,  subtract  drafts,  because  the  former  come  to  me,  the 
latter  go  from  me  ;  opposite  directions  as  before. 

We  might  also  give  illustrations  involving  time  past  and 
future,  suggestive  of  direction  backwards  and  forwards,  as 
well  as  other  varieties  of  oppositeness,  not  in  the  nature  of  the 
quantity,  but  in  its  relation  to  the  problem;  all  of  which, 
without  severe  stretch  of  the  imagination,  may  be  called 
opposite  directions. 

82.  These  opposite  directions  determine  whether  a  quan- 
tity shall  be  added  or  subtracted ;  that  is,  they  determine  the 
direction  in  which  the  quantity  is  to  be  used  ;  for  addition  and 
subtraction,  by  which  a  quantity  is  put  in  or  taken  out  as  a 
term,  may  be  considered  as  opposite  directions. 

83.  It  will  be  observed  that  a  quantity  having  a  particular 
direction  is  not  always  to  be  added,  nor  is  one  having  the  oppo- 
site direction  always  to  be  subtracted ;  but  when  the  conditions 
require  one  to  be  added,  the  other  must  be  subtracted. 

84.  These  opposite  directions  are  called  Positive  and 
Negative  f  that  direction  which  tends  to  increase  the  quan- 
tity sought  being  usually  called  positive,  and  the  opposite 
direction  negative,  though  either  direction  may  be  assumed  as 
positive  at  pleasure. 

85.  Since  the  signs  -f-  and  —  represent  the  direction  of  a 
quantity,  they  may  be  called  Directive  Siyiis,  or  Fac- 
tors of  Direction. 


24  NOTATION. 

86.  In  the  use  of  these  signs  it  is  necessary  to  consider  the 
following 

PRINCIPLES. 

i°.  A  quantity  may  be  considered  without  reference  to  its 

direction.     It  has  then  no  sign,  and  is  neither  positive  nor 
negative. 

Thus,  the  answers  to  the  questions,  How  far?  How  long?  How 
many  ?  etc  ,  are  of  tins  nature. 

Bu1  when  a  question  includes  the  idea  of  direction,  as,  How  far 
north?  How  Ion.;-  before?  How  long  after?  How  much  did  you 
receiti  .'  etc.,  the  answer  must  be  either  +  or  — . 

2°.  Questions  involving  the  idea  of  direction  may  always 
be  reversed. 

Thus,  How  in  north?  How  far  south  ?  How  much  did  you  receive? 
How  much  did  yon  give?  etc. 

3°.  The  nature  of  problems  is  such,  that  tie  quantities 
involved  must  //(/re  one  of  two  opposite  directions. 

When  a  problem  asks,  "How  far  north  or  south?"  other 
directions  have  nothing  to  do  with  the  answer.  So  of  time 
past  or  future.  Mils  payable  and  bills  receivable,  etc...  the  only 
possible  direct  inns  are  two,  and  these  are  opposite. 

NOTE. — Nothing  but  some  impossible  condition  in  a  problem  will 
introduce  into  the  solution  a  quantity  out  of  the  line  of  positivt  and  nega- 
tie  ,  and  wben  by  reason  of  an  impossible  condition  such  a  quantity  is 
introduced,  it  is  called  an  impossibl  of  lino, jinn ry  quantity,  and  its 
direction  is  expressed  by  a  method  to  be  explained  hereafter  (Art.  292.) 

4°.   There  is  no  direction  which  is  naturally  positive,  but  it 

is  cum  1  unary  to  consider  the  direct  inn  named  in  a  problem  as 
positive,  unless  the  opposite  be  made  so  by  special  assumption. 

87.  In  operations  upon  quantities  having  directive  signs,  it 
is  no!  necessary  to  consider  the  nature  o£  the  quantities,  nor 
bhe  hind  of  oppositeness  which  gave  rise  to  these  signs,  whether 
of  distance  or  time,  or  of  debt  and  credit  ;  for  when  once  the 
equations  arc  formed,  the  signs  are  used  in  accordance  with 
the  arbitrary  meaning  assigned  to  them  ;  chat  is.  in  accordance 
with  the  following  definitions  of  the  directive  signs. 


NOTATION.  25 


DIRECTIVE      SIGNS. 

88.  The  sign  -\-  indicates  the  positive  direction,  but  has 
no  power  to  control  the  direction  of  a  quantity  in  the  presence 
of  the  sign  — . 

89.  The  sign  —  reverses  the  direction  of  a  quantity. 

90.  In  the  application  of  these  definitions  we  observe : 
ist.  The  sign  +  having  no  power  as  a  factor  of  direction, 

may  be  omitted,  except  when  its  omission  would  lead  to  some 
misunderstanding;  jnst  as  the  factor  i  is  omitted  because  it 
has  no  power  as  a  factor  of  magnitude  or  value. 

2d.  Two  or  more  +  signs  mean  no  more  than  one,  as  two 
or  more  i's  as  factors  have  no  more  effect  than  one  such 
factor. 

3d.  When  a  quantity  has  several  signs,  some  of  which  are 
+  and  some  — ,  the  direction  of  the  quantity  will  depend 
wholly  on  the  negative  signs. 

4th.  When  a  quantity  has  several  minus  signs,  each  of  these 
signs  will  inverse  its  direction.     For  illustration  : 

Let  the  minute-hand  of  a  clock,  when  pointing  to  the  hour 
XII,  represent  the  positive  direction  of  a  quantity,  or  its 
direction  with  no  written  sign. 

A  single  minus  sign  will  reverse  that  direction,  turning  the 
hand  backwards*  till-it  jwints  to  VI. 

Another  minus  sign  will  continue  this  revolution,  and 
bring  the  hand  back  to  XII,  or  the  positive  direction.     Thus 

we  see  that  —  a  is  negative, a  is  positive,  and a 

is  negative,  etc.  These  may  therefore  be  written  —  a,  +  a, 
—  a,  etc.,  or  if  we  wish  to  show  how  many  minus  signs  are 
used  in  giving  the  direction,  we  may  use  an  exponent  for  this 
factor  of  direction,  as  well  as  for  factors  of  magnitude,  and 
write,   — a,  — 2a,   — 3a,  — ia,  etc. 

*  We  say  backward*  (meaning  opposite  to  the  natural  motion  of  the  hands  of  the 
clock),  because  it  is  customary  to  consider  revolution  in  this  direction  as  pod/ire,  and 
in  the  opposite  direction  negative.  Taking  away  a  —  sign  from  a  quantity  would 
reverse  it  in  the  negative  direction.  There  is  nothing,  however,  to  forbid  reversing 
this  supposition  and  making  positive  revolution  agree  with  the  natural  motion  of  the 
hands. 

2 


26  NOTATION. 

91.    From  these  illustrations  we  have  the  following 

Rule.— An  even  power  of  —   is  positive,  an  odd  power 
negative. 

Or,  An  even  number  of  minus  signs  gives  +  ,  and  an  odd 

a  a  in  her  — . 

92.   Apply  this  rule  to  the  following 


EXAM  PLES. 

i.  What  is  the  sign  of  -\ \-  a? 

2.  What  is  the  sign  of  — 2 1 2a? 

3.  What  is  the  sign  of  a  and  what  of  b  in  the  following 
expression  : \-  (a  —  b)  ? 

4.  What  is  the  sign  of  —  ±  +  a?  Ans.   zf. 

5.  What  is  the  sign  of ±a?  Ans.   +  . 

6.  What  is  the  sign  of  ±  —  =F  «  ?  Ans.    -f- . 

Note. — When  there  are  several  double  signs,  the  upper  signs  are 
generally  taken  together,  and  also  the  lower.  Thus,  in  the  last  exam- 
ple, the  sign  of  a  will  be  either   + or + ,  each  of  which  will 

give   +. 

93.   What  signs  should  be  given  to  the  following  answers  ? 

7.  How  far  north  did  you  go  ?  Ans.   10  miles. 

8.  How  much  did  you  pay?  Ans.   5  dollars. 

9.  How  much  older  are  you  than  John  ? 

Ans.  5  years  younger. 

10.  How  much  older  than  Henry?  Atis.  3  years. 

11.  How  many  books  have  you  ?  Ans.  10. 

12.  How  many  hooks  did  you  take  from  the  table? 

Ans.  5. 

13.  How  old  are  you  ?  Ans.  50  years. 

14.  How  far  is  it  to  New  York?  A?is.  100  miles. 

Some  of  the  above  answers  have  no  sign;  which  are  they, 
ami  why ? 


NOTATION.  27 

94.  The  force  of  a  directive  sign  is  limited  to  the  term 
immediately  following,  unless  several  terms  are  connected  by  a 
parenthesis  or  vinculum. 

Thus,  in  —a?b  +  ab'2,  the  sign  —  affects  only  a%;  but  if  we  write 
—  (a26  +  ah2),  the  whole  quantity  is  negative. 

95.  In  like  manner,  the  sign  —  before  a  fraction ;  as, 
,  affects  the  whole  fraction,  and  if  in  the  course  of  a 


2 

solution  the  denominator  be  removed,  it  must  not  be  written 
—  a  —  b,  but  —  (a  —  b),  or  —  a  +  b. 


EXERCISES     IN     NOTATION. 

96.  To  Translate  an  Algebraic  Statement  from  Common  into 
Algebraic  Language. 

i.  The  product  of  the  sum  and  difference  of  any  two  quan- 
tities is  equal  to  the  difference  of  their  squares. 

Solution. — In  this  translation,  it  is  necessary  first  to  assume  letters 
to  represent  the  quantities.  Let  a  and  b  be  the  letters.  Then  the 
statement  becomes  (a  +  b)(a  —  b)  =  a-  —  b-,  Ana.     Hence,  the 

Eule. — For  the  words,  substitute  the  letters  and  signs 
which  indicate  the  relations  of  the  quantities  and  the  operations 
to  be  performed. 

Translate  the  following  into  algebraic  language : 

2.  The  square  of  the  sum  of  any  two  quantities  is  equal  to 
the  sum  of  their  squares  increased  by  twice  their  product. 

3.  The  square  of  the  difference  of  any  two  quantities  is 
equal  to  the  sum  of  their  squares  decreased  by  twice  their 
product. 

4.  The  square  of  the  sum  of  any  two  quantities  added  to 
the  square  of  their  difference  is  equal  to  twice  the  sum  of 
their  squares. 

5.  The  difference  of  the  squares  of  two  quantities  divided 
by  the  difference  of  the  quantities  equals  the  sum  of  the 
quantities. 


28  NOTATION. 

6.  The  difference  of  the  square  roots  of  two  quantities 
divided  by  the  sum  of  their  fourth  roots  equals  the  difference 
of  i  licit-  fourth  roots. 

7.  The  square  of  the  difference  of  two  quantities  subtracted 
from  the  square  of  their  sum  is  equal  to  four  times  their 
product. 

97.  To  Translate  Algebraic  into  Common  Language. 

1.  Translate  (x  +  y)2  —  (x  —  y)2  =  Ley  into  common 
language. 

Solution. — The  square  of  the  sum  of  any  two  quantities,  diminished 
tiy  the  square  of  their  difference,  is  equal  to  4  times  their  product 
Hence,  the 

Rule. — For  the  letters  representing  quantities  and  the  sigm 
indicating  the  given  relations  and  operations,  substitute  words. 

Translate  into  common  language  the  following 

2.  (a  +  b)3  =  a3  -f-  yi2b  +  yib2  +  b3. 

3.  1  («  +  i)  +  1  {a  —  h)=a. 
4-     i  (a  +  b)  -  i  (a  -  b)  =  b. 

98.  If  a  =  3,  b  =  2,  c  =  |,  m  =  a  +  b,  n  =  a  —  b,  what 
are  the  numerical  values  of  the  following  expressions: 

l(a-b)(a  +  b) 
5'  <? 

6.  2a2  +  b2  —  2  (a2  —  b2)  —  2  {inn  —  1). 

7.  (tf-y-*-^1)*. 

S.     Vc2  [{a  +  bf  -a2]. 


9.     a2  +  b2  —  (a  -  b)2  • . 

ta±b_a-b\thc  _jn  n\ 
v     //  m    I  \  a  -\-b' 


10. 


a  +  bl 

,,W v-2  —  (m2  +  n2)  b2c  +  ^c2" 


Nora— Let  the  teacher  add  examples  until  the  student  becomes 
familial  with  algebraic  language. 


CHAPTER    I  I. 

ADDITION. 

99.  Algebraic  Addition  is  uniting  the  terms  of  two 
or  more  quantities  in  one  expression,  and  reducing  that  expres- 
sion to  the  simplest  form. 

100.  The  result  is  called  the  Sum  or  Amount. 

101.  Algebraic  addition  depends  upon  the  following 

PRINCIPLES. 

i°.  Similar  terms  only  may  be  united  in  one.     (Art.  78.) 

20.  Dissimilar  terms  can  only  be  connected  by  their  signs. 
(Art.  79.) 

102.  Algebraic  addition  differs  from  Arithmetical  in  the 
fact  that  the  quantities  added  may  be  either  positive  or  nega- 
tive. In  Arithmetic,  the  signs  +  and  —  are  used  merely  to 
express  addition  and  subtraction,  and  quantities,  whether 
added  or  subtracted,  are  used  as  positive. 

103.  In  Algebra,  for  convenience,  we  speak  of  finding  the 
aggregate  of  terms  of  both  kinds,  positive  and  negative. 

A  man,  for  example,  rows  a  boat  up  stream  while  the 
current  floats  it  down.  The  first  hour  he  rows  3  miles  and  is 
floated  down  1  mile ;  the  second  hour  he  rows  4  miles  and  is 
floated  down  2  miles ;  the  third  hour  he  rows  2  miles  and  is 
floated  down  2  miles. 

Now  to  find  the  aggregate  effect  of  the  oars  and  current  is 
a  problem  for  Algebraic  Addition. 


30  ADDITION. 

In  Arithmetic,  it  would  be  called  in  part  subtraction, 
as  ii  really  is,  and  the  process  of  aggregating  the  quantities  in 
Algebra  does  nut  differ  from  that  which  would  be  employed  in 
Arithmetic. 

Calling  the  distances  the  man  rows  positive,  the  distances 
he  is  floated,  being  in  the  opposite  direction,  would  be  negative, 
and  the  aggregate  effect  is  evidently  the  difference  of  the  posi- 
tive and  negative  distances,  which  in  this  case  is  9—5,  or 
4  miles. 

104.  The  relation,  however,  of  rowing  and  current  might 
be  such  as  "to  make  the  result  greater  or  less  than  this.  The 
current  might  even  be  more  rapid  than  the  rowing,  and  the 
boat  he  found  farther  down  stream  than  when  it  started. 

This  would  be  shown  by  the  negative  distances  being 
greater  than  the  positive,  giving  a  negative  sum. 

Note. — This  illustration  shows  the  meaning  of  the  remark, "  A  nega- 
tive quantity  is  less  than  zero."  Not  that  a  negative  distance  is  longer  or 
shorter  than  it  would  be  if  positive,  but  when  a  man  wishes  to  go  up 
stream,  it  is  worse  than  nothing  to  him  to  be  floated  down. 

105.  Hence  we  see  that 

Adding  a  negative  quantity  is  equivalent  to  subtracting  a 
positive  one  of  the  same  numerical  value,  and  vice  versa. 

Adding  the  effect  of  the  current  is  the  same  in  its  result  as 
taking  away  an  equivalent  rowing  effect. 

106.  The  Algebraic  Sum  of  several  quantities  is  the 
difference  between  the  sum  of  the  positive  and  the  sum  of  the 
negative  quantifies,  with  the  sign  of  the  greater  sum.    Hence, 

107.  For  Algebraic  Addition  we  have  the  following 

GENERAL  RULE. 

I.  Unite  similar  terms  by  prefixing  the  algebraic  sum  of  the 
coefficients  to  tin-  oommon  letters.     (Art.  46.) 

II.  Connect  dissimilar  terms  by  their  signs.     (Art.  45.) 


ADDITION.  31 

108.  Terms  that  are  similar  in  respect  to  one  or  more 
letters  may  be  united  with  a  polynomial  coefficient. 

Unite  2abc  +  $abx  —  7,aby  +  2abn  with  a  polynomial 
coefficient. 

Solution. — The  common  letters  are  ah.     The  sum  of  the  coefficients 
is  2C  +  5CC— 3y  +  2ii,  and  we  have  by  the  rule, 
(2c  +  501— yy  +  in)  ah. 

109.  In  such  cases,  the  coefficient  of  each  term  must  be 
regarded  as  containing  all  the  factors  of  the  term  except  the 
common  letters. 

These  coefficients  therefore  will  sometimes  be  literal,  in 
which  case  their  sum  can  be  expressed  only  by  a  polynomial 
as  above.  Hence  it  will  be  a  matter  for  consideration  whether 
the  simplified  expression  will  be  more  convenient  for  use  than 
the  full  form. 

EXAMPLES. 

i.  Add  2ab  +  $cd  —  &d;  $cd  —  ab  +  2d\  \ax  -\- $ab -{- 2d ; 
\d  —  3&C  —  2ctx. 

Solution. — For  convenience,  we  write  similar  terms  under  each 
other ;  thus* 

2ah  +  5«Z  —  M  +  <\ax  —  36c 
—    ab  +  yd  +  2d  —  2ax 
+  $ab  +  2d 

+  yd __ 

bah  +  8cd  +  2ax  —  36c,  Ans. 

2.  Add  yt  -f-  7X  5  2a  —  5^5  3X  —  2($\  ax  +  5 a  )  3X  +  a- 

3.  Add  zofl  +  yvi  ;  2ub-  —  a*;  a*  —  ab. 

4.  Add  5 a3 J  -|-  Tab2 ;  —  7  ax2  +  dlb ;  —  ax2  4-  ytb2  —  a2x. 

5.  Add  ytb2  —  4«2/;  4-  a3;  sab2  —  <\ac2  —  cs;  2a2b  —  jab2 
—  6ae. 

6.  AdA  4XJJ2  4-  4.T3?/;  $x2y  4-  2xy2;  jx2y2  —  $xy\ 

7.  Simplify  ax  —  bx  4-  2ax  —  bx  4-  ex  —  2ax. 

8.  Simplify  a^b?  4-  ab-  —  a*b  4-  20*$  —  2ab*. 

9.  Simplify  axif  4-  200^  —  3«.y2^  —  x2«y. 


32  SUBTRACTION. 

10.  Simplify  3a/«^  +  2  Vac  —  2  (ad)1-  —  «M. 

11.  Simplify  tab  +  ^a2b  —  $ab  +  zb. 

12.  Simplify  3./^  -|-  2ay-  —  3%2. 

13.  Simplify  ran  —  zan  +  5». 

110.   Sim Ihi)'  Polynomial  Terms  maybe  united 
as  well  as  monomials ;  as, 

14.  Add  2  (x  +  y)  +  3  (?  —  y) ;  3  («  +  #)  —  (3  -  y)- 

Ans.  5  (a;  +  y)  +  2  (.7;  -  ?/). 

15.  Add  2  (a  +  5)*  4-  3\/rt  —~b  —  Va  +  6  —  4  («  —  5)i 

16.  Add  «V^  —  1  ;  —  b's/x  —  1 ;  —  c  (a;  —  1)*. 

17.  Add  2  (a  —  xp  —  4V a  —  x  +  3  (a  —  #)*. 


SUBTRACTION. 

111.  Subtraction  is  taking  one  quantity  from  another. 
The  Minuend  is  the  quantity  from  which  the  subtrac- 
tion is  made. 

The  Subtrahend  is  the  quantity  subtracted. 
The  Difference  is  the  result  of  subtraction. 

112.  Algebraic  subtraction  depends  upon  the  following 

PRINCIPLES. 

i°.  Similar  quantities  only  can  be  subtracted  one  from 
another. 

20.  Subtracting  a  positive  quantity  is  equivalent  to  adding 
an  equal  negative  one. 

3°.  Subtracting  >>  negative  quantify  is  the  *an@  a*  adding 
an  equal  positive  one. 

4°.  The  sum  of  the  difference  and  subtrahend  is  equal  to  the 
minuend. 


SUBTRACTION.  33 

113.  The  only  difference  between  Subtraction  and 
Addition  in  Algebra  is  that  when  we  propose  to  subtract  a 
quantity,  we  give  that  quantity  another  negative  sign,  and 
therefore  change  all  its  signs.     Hence  the  following 

GENERAL      RULE. 

114.  Change  the  signs  of  quantities  to  be  subtracted,  and 
unite  the  terms  as  in  addition.     (Art.  107.) 

Notes. — 1.  For  convenience,  similar  terms  may  be  written  under 
each  other,  as  in  addition.     (Art.  109.) 

2.  When  a  result  is  to  be  found  from  several  quantities  by  adding 
some  and  subtracting  others,  the  operation  may  be  made  one  by  the 
above  rule. 

3.  In  finding  the  difference  between  two  quantities,  it  is  immaterial 
which  is  made  the  subtrahend.  The  result  will  be  the  same  in  either 
case,  except  its  sign. 

Thus,  the  difference  between  7  and  4  is  7  —  4  =  3,  or  4  —  7  =  —  3. 

When,  therefore,  a  problem  requires  only  the  difference,  without 
regard  to  sign,  either  quantity  may  be  made  the  subtrahend. 

EXAMPLES. 

1.  From  a2  +  2ax  +  x2  take  a2  —  2ax  +  x2. 

2.  From  $x2  —  2^  +  5  take  x2  —  2  +  x. 

3.  From  $ab  +  b2  take  —  2ab  +  b2. 

4.  From  a2  +  2ax  -+-  x2  take  ax  —  a2  +  x2. 

5.  From  a  (x  +  y)  —  b  (x  +  y)  take  (x  +  y)  —  b  (x—y). 

6.  From  3\/#2  +  «2  take  5V%2  +  «2. 

7.  From  a  +  b  —  c  +  d  take  a  —  b  +  c  +  d. 

Simplify  the  following: 

8.  3^A_  [(«  _|_  l&  —  tf)  _  (zx3  +  zbx2  +  a)]. 

9.  3  (ax2  —  ab2)  —  3a  (x2  —  b2). 
10.     (nix)?  +  m*z*  —  2\/'mx. 


34  SUBTRACTION. 

ii.     2db  —  ib~  +  ib  (a  +  b). 

12.  a  +  b  —  c  —  //>  -f  »  —  (a  —  &  +  c  +  m  +  n). 

13.  2«i  —  /*«  —  («b  +  &2). 

USE     OF     PARENTHESES. 

115.  It  will  be  observed  in  the  above  examples  that  the 
use  of  Ptirrntlieses  is 

1  st.  To  bring  several  terms  under  the  influence  of  the 
sign  -. 

2d.  To  connect  several  terms  to  the  same  coefficient. 
3d.  To  subject  several  factors  to  the  same  exponent. 

116.  These  parentheses  may  be  removed  in  the  several 
cases  : 

1st.  By  applying  the  sign  —  to  each  term  separately,  or, 
what  is  the  same  thing,  by  changing  the  sign  of  each  term. 
2d.   By  connecting  the  coefficient  with  each  term. 
3d.   By  applying  the  exponent  to  each  factor. 

117.  The  parenthesis  or  vinculum,  when  used  to  save  the 
repetition  of  some  sign,  coefficient,  or  exponent,  ca^i  generally 
be  removed  without  changing  the  value  of  the  expression: 

By  applying  the.  sign,  coefficient,  or  exponent  to  each  of  the 
quantities  separately  to  which  it  belongs. 

118.  In  doing  this,  it  must  be  remembered  that  coefficients 
belong  to  terms,  and  exponents  to  factors.  We  nausl  not 
therefore  apply  an  exponent  to  the  several  terms  nor  a  coef- 
ficient to  the  several  factors  of  the  quantity  within  the 
parenthesis. 

Thus,  the  expression  2  (fib  +  b")  does  not  equal  2a-  26  +  26s  (applying 
the  2  to  both  factors  a  and  6),  but  211b  +  2b'2. 

So  (a  —  /')-  is  not  the  same  as  aa  —  ft4.  « 

Note. — The  parenthesis  is  here  used  to  indicate  that  a  —  b  is  used 
as  a  single  factor,  and  it  cannot  be  removed  in  the  manner  explained 
above. 


SUBTRACTION- 


35 


119.   Remove  the  signs  of  abbreviation  from  the  follow- 
ing, and  reduce  to  the  simplest  form : 

i.     2  \a  -  [b  +  (c  +  zx)  —  (y  —  2)]}. 

2.  3  [a  —  (b  +  c)  +  2  (to  —  i)  —  (to  +  i)]. 

3.  (a  +  b  —  c)  —  (b  —  a  +  c). 


a\x  —  a  I  x  +  2a\x. 


—  c 


—  c\     +    b 


5.  ^abc'  —  2\/abc  +  {2abc)^. 

6.  a  [a  —  a  (b  —  c)]  —  (a2  +  «2Z>)  +  a?c. 

7.  Vab  +  d*b$  —  yxif  —  2(a%  fyb2  +  2a;?/2  —  aW). 

8.  x—s\[d$+  tab  —  (ab)i] 

_  \ab  +  a*j£  -  (^p)"3  -x)]\, 

9.  (a  +  5  —  c)  V%  +  #  —  (a  +  *  +  c)  (jb  +  y)i 

10.  (a  +  2«^to  +  5)  —  (a  —  2\/ab  +  b). 

11.  (a  —  2«2#2  4-  j)  _2  (^  _j_  2\/aVb  ±  #)• 

12.  ?w  (a  4-  b)  —  wi  («  —  &)  +  2m  (J  —  a). 


120.  In  the  present  chapter  we  have  considered  quantities 
as  Terms.  It  now  remains  to  treat  of  them  as  Factors,  in 
connection  with  the  following  subjects,  viz.  : 

Multiplication,  Division,  Factoring,  Greatest  Common 
Divisors,  Least  Common  Multiples,  Fractions,  Powers,  and 
Roots. 

7v 


CHAPTER    III. 

MULTIPLICATION. 

121.  Multiplication  is  the  process  by  which  the  sum 
of  any  number  of  equal  terms  is  found  without  performing  the 
addition. 

The  Multiplicand  is  the  quantity  multiplied. 

The  Multiplier  is  the  number  by  which  it  is  multiplied. 

The  Product  is  the  result  of  multiplication. 

The  multiplier  and  multiplicand  are  called  Factors. 

122.  To  illustrate  the  difference  between  multiplication 
and  addition,  take  the  following  example  : 

What  is  the  sum  of  four  distances,  each  equal  to  5  feet? 

We  may  find  the  answer  to  this  by  saying  mentally,  5  and  5  are  io, 
and  5  are  15,  and  5  are  20;  or  we  may  say.  4  times  5  are  20.  The  men- 
tal work  of  the  first  is  that  of  successive  additions ;  of  the  second  it 
consists  simply  in  giving  the  required  sum  from  memory,  knowing  the 
/,  mi  and  number  of  times  it  is  used. 

In  addition,  if  one  does  not  remember  what  numbers  and  5  make,  he 
may  begin  at  5  and  count,  adding  one  at  a  time,  and  thus  find  the  sum. 

In  multiplication,  the  result  can  be  given  only  by  memory,  which 
associates  certain  products  with  certain  factors,  and  if  the  memory  fail, 
the  answer  can  be  found  only  by  addition. 

In  the  example  above,  when  the  20  is  considered  as  made  up  by  the 
addition  of  four  lives,  it  is  called  a  sum  :  but  when  it  is  found  by  con- 
sidering the  numbers  5  and  4,  and  giving  from  memory  the  result  without 
adding,  it  is  called  >t  /induct. 

The  two  operations  ;,re  expressed  thus  : 

1st.  5  +  5-1-5  +  5  =  20. 

2d.  5x4   =   2D. 

In  the  first,  the  5's  are  terms  and  the  20  is  their  sum.  In  the  second, 
5  and  4  are  factors  and  20  is  their  product,     (Art.  11.) 


MULTIPLICATION.  37 

Note. — It  is  important  that  the  distinction  between  terms  and  fac- 
tors should  be  kept  in  mind : 

ist.   Terms  are  quantities  united  by  algebraic  addition. 

2d.  Factors  are  quantities  one  of  which  shows  how  many 
times  the  other  is  used  as  a  term. 

123.  A  product  formed  by  two  factors  may  be  multiplied 
by  a  third  factor,  and  this  product  by  a  fourth,  and  so  on 
indefinitely.  A  product  may  therefore  contain  any  number  of 
factors.     Hence  the  definition  : 

124.  M/ultiplication  is  the  process  of  combining  fac- 
tors into  a  product, 

PRINCIPLES. 

125.  i°.  Tlie  multiplier  must  be  considered  an  abstract 
quantity. 

2°.  The  product  is  of  the  same  nature  as  the  multiplicand ; 
for,  repeating  a  quantity  does  not  alter  its  nature. 

3°.  The  product  of  two  or  more  factors  is  the  same  in 
whatever  order  they  are  multiplied. 

126.  Multiplication  may  be  considered  under  two  cases : 
ist.  Multiplication  of  monomials. 

2d.    Multiplication  of  polynomials. 

CASE      I. 

127.  Multiplication  of  Monomials. 

By  algebraic  notation  (Art.  53),  we  have  for  the  multipli- 
cation of  monomials  the  following 

Rule. — To  the  product  of  the  numerical  coefficients  annex 
the  literal  factors,  giving  each  an  exponent  equal  to  the  sum  of 
all  its  exponents  in  the  several  factors. 

Note. — This  rule  applies  to  any  number  of  monomial  factors. 

128    The  Signs  of  the  Factors  in  multiplication  are 

to  be  treated  as  factors  of  direction,  and  must  enter  the  product 
precisely  as  the  other  factors.     (Art.  91.)     Hence, 


38  MULTIPLICATION. 

129.  For  the  Siyus  in  Multiplication  we  have  the 
following 

IiUle. — An  even  power  of  —  is  positive}  an  odd 
power  negative. 

Or,  An  err//  number  of  negative  factors  gives  a  positive 
product,  an  odd  number  a  negative. 

130.  When  there  are  only  two  factors,  this  gives  the 
common 

Eule. — Like  signs  give  +  ;  unlike,  — . 

EXAMPLES. 
Find  the  product  of  the  following : 

i.     2«b2.c  x  3«2£3£2.  Ans.  6asb5z?. 

2.     yfixy  x  zaxy%  x  sa3x-  x  \<ibc.   Ans.  \c)a'iucxiyz. 
3-     2i(v-Ay"  x  %z*yz  x  a?z. 

,      3,2  .       2  7  :!  T      6  ,  I     „ 

4.  %aw3c  x  %a3b*z  x  ^c^b^x2. 

5.  "]a2x\)f  x  3f/^2//  x  rtto. 

6.  $ambnz  x  2«^2i  x  |«&. 

7.  2an&m  x  3c«OT//ra  x  aW~. 

8.  15  a1-^1""  x  |rtm^  x  ax. 

9.  1  art1"™//1-'1  x  |aP_1Jre_1. 

11.  flin+nJjm-71   v   ((m-nfom+n^ 

12.  (ab)m"n  x  am-nbm-Jl. 

13.  xm~1yn~®  x  :n/2. 

14.  x*>-^yn   x  .r"//1   ". 

1  c .  a;72,3  wm:!  x  #"  (w— 2)  f/wj  (m— l)^ 

16.  2rt2.'-  x  —  3SC5  x  —  rt.r2  x  a5. 

17.  rt52  x   ±  rtV  x  —  frr2  x  rt//r. 

18.  -\/a£   x   —  (>"Ir-   X  (—  rt.r)2  x  2rt#. 

19.  $fyax  X   —  2\/rt.r  x  {—a '■•'"-')   X  rt.'C2. 

20.  (—  rt.r3)  (—  ""'./)  (,/V2)  y/ax. 


MULTIPLICATION.  39 


CASE      II. 
131.    Multiplication  of  Polynomials. 

The  Multiplication  of  Polynomials  is  performed 
by  the  following 

Rule. — Multiply  each  term  of  the  multiplicand  by  each  term 
of  the  multiplier,  and  add  the  products. 

Notes. — i.  This  does  not  differ  in  principle  from  the  method  of 
multiplying  numbers,  where  each  figure  is  multiplied  separately  and  the 
products  added.     The  multiplier  may  be  a  monomial, 

2.  For  convenience  in  adding  the  partial  products,  like  terms  should 
be  placed  under  each  other. 

3.  The  multiplication  of  polynomials  may  be  indicated  by  inclosing 
each  factor  in  a  parenthesis,  and  writing  one  after  the  other.  Thus, 
{a  +  b  +  c)  (a  +  b  +  c )  is  equivalent  to  (a  +  b  +  c)  x  (a  +  b  +  c). 

21.   Multiply  x2  —  2ax  -f-  a2  by  a  +  x. 

Solution. — We  write  the  multiplier  under  the  multiplicand,  and 
proceed  thus  : 

OPERATION. 

x2  —  iax  +  a? 
a  +  x 


Multiplying  by  a,  ax'1  —  2<i'-x  +  of 

Multiplying  by  x,  —  2ax2  +    a2x  +  xz 


Adding  products,  —    ax1  —    a^x  +  a'6  +  x*,  Ann. 

Multiply  the  following: 

22.  (a2  +  2ax  +  x~)  (a2  —  2cix  +  x2). 

23.  20b  (a2  —  ax2  —  b2). 

24.  (a  +  x)  (a2  -f  a2)  (a2  —  x2). 

25.  (2ax  —  x2)  (a  -f-  bx  +  ex2). 

26.  (2a  +  bx  —  ex2)  (2a  —  bx  +  ex2). 

27.  (am  -f  bn)  (am  +  b")  (am  —  bn). 

28.  (1  +  x)  (1  —  x)  (1+*-  x2)  (1  —  x  +  x2). 


40  MULTIPLICATION. 


MULTIPLICATION    BY    DETACHED    COEFFICIENTS. 

132.  When  the  terms  of  the  multiplicand  and  multiplier 
can  both  be  arranged  so  that  the  exponents  of  each  letter 
increase  or  decrease  by  unity  in  the  successive  terms,  we  may 
write  simply  the  coefficients,  and  afterwards  introduce  the 
letters  into  the  product. 

29.  Multiply  a2  -f  2  ax  —  x2  by  a  +  x,  by  detached 
coefficients. 

Solution. — We  write      1  +  2  —  1 
1  +  1 


1  +  2  —  1 

i_+  .2  —  1 

1  +  3  +  1  —  1 
a3  +  3a?x  +  art2  —  x3,  Product. 
By  comparing  this  with  the  multiplication  of  the  same  quantities  by 
the  ordinary  method,  the  process  will  be  made  plain. 

133.  By  this  method  of  detached  coefficients,  a  term 
must  be  introduced  with  o  for  its  coefficient  when  necessary 
to  make  the  exponents  change  by  unity  in  the  successive  terms. 

For  example,  to  multiply  a3  +  2ax°— x3  by  a-  —  x-,  we  must  introduce 
a  term  between  the  first  and  second  in  each  factor ;  thus, 

1  +  o  +  2  —  1 

I   +  O  —  I 

I +0+2—1 

—  I—  O—  2   +   1 


I+O+l  —   I—  2   +    1 

Product,    ah  +  oa4;r  +  aW—tfx3— 2ax*  +  x5, 
from  which  the  second  term,  having  o  for  a  coefficient,  may  be  omitted. 

EXAMPLES. 
Find  the  product  of  the  following: 

30.  (3a5  —  2aib  +  jaW  —  56s)  (d*b  —  aW). 

31.  (a3  —  2ax2  +  x3)  (a2  —  a;2). 

32.  (tT4    _    yi)    (tf    +    ^\- 

33-     K  -  5«?^  +  3a*5  -  a*)  {a2  —  ax). 


CHAPTER    IV. 

DIVISION. 

134.  Division  is  finding  how  many  times  one  quantity 
is  contained  in  another;  or,  finding  the  measure  of  a 
quantity  with  a  given  quantity  as  a  unit  of  measure. 

The  Dividend  is  the  quantity  to  he  divided,  or  measured. 

The  Divisor  is  the  quantity  by  which  we  divide,  or  the 
unit  of  measure. 

The  Quotient  is  the  number  found  by  division. 

135.  Division  is  the  reverse  of  Multiplication, 

the  dividend  being  the  product,  the  divisor  and  quotient  the 
factors. 

Multiplication  combines  factors  into  a  product ;  Division 
removes  factors  from  &  product. 

Multiplication  finds  the  sum  of  a  $wew  number  of  e^wrt? 
terms;  Division  finds  Ao?tf  w?a«?/  tfzmes  a  ^wew  term  must  be 
taken,  or  what  term  must  be  taken  a  given  number  of  times,  to 
produce  a  given  sum. 

136.  The  relations  of  Dividend,  Divisor,  and  Quotient 
give  us  readily  the  following 

PRINCIPLES. 

i°.  Multiplying  or  dividing  the  dividend  ?nultiplies  or 
divides  the  quotient. 

2°.  Multiplying  or  dividing  the  divisor  divides  or  multiplies 
the  quotient. 

3°.  Multiplying  or  dividing  both  dividend  and  divisor  by 
the  same  factor  does  not  affect  the  quotient. 


42  DIVISION. 

137.  We  have  three  cases  in  Division: 

i st.  Dividing  a  monomial  by  a  monomial. 
2d.    Dividing  ^polynomial  by  a  monomial. 
3d.   Dividing  &  polynomial  by  a  polynomial. 

CASE      I. 

138.  To  Divide  a  Monomial  by  a  Monomial. 

The  Division  of  Monomials  is  performed  by  the 
following 

Kule. — Divide  the  numerical  coefficient  of  the  dividend  by 
that  of  the  divisor,  and  annex  to  the  quotient  the  letters  of  both 
dividend  and  divisor,  giving  each  an  exponent  found  by 
subtracting  its  exponent  in  the  divisor  from  its  exponent  in  the 
dividend.  ' 

Note. — The  reason  of  this  rule  is  plain.  Since  the  factors  of  the 
divisor  are  to  be  taken  from  the  dividend,  it  is  evident  that  any  factor 
will  be  found  in  the  quotient  as  many  times  as  it  is  found  in  the  dividend 
minus  the  number  of  times  it  is  found  in  the  divisor. 

EXAMPLES. 

1.  Divide  Sa^b^x4  by  2acx'i. 

Solution. — Dividing  the  coefficient  8  by  2,  we  have  4,  to  which  we 
annex  the  letters;  thus,  ^abcx.  To  find  the  exponents  of  these  letters, 
we  have  for  a,  2  —  1  =  1 ;  for  b,  1  —  0=1;  for  c,  3  —  1  =  2;  for  x, 
4  —  2  =  2.  Applying  these  exponents,  and  remembering  that  the  expo- 
nent 1  need  not  be  written,  we  have  as  the  quotient  ^abc-x?. 

2.  Divide  jaW&v3  by  arb&x. 

Solution. — Following  the  same  order,  we  find  jahc°x-.  Here  we 
have  0  for  tli''  exponent  of  r,  showing  that  the  factor  c  is  not  found  in  the 
quotient.     We  may  therefore  omit  it  and  write,  jabx1*. 

3.  Di\ ide  5a2  by  5a3. 

Solution. — Dividing  as  before,  we  get  1  for  the  coefficient  and  zero 
for  the  exponent  of  a  in  the  quotient.  The  quotient  is  therefore  1,  as  it 
should  be,  since  the  dividend  and  divisor  are  the  same.  If  we  choose, 
we  may  write  ■'"  and  omit  the  coefficient  1. 

NOTE. — It  follows  that  any  quantity  with  o  for  an  exponent  is  equal  to  1. 


DIVISION.  43 

4.  Divide  4a2 be3  by  2aUlc. 

Solution.— Following  the  rule  as  before,  we  find  that  the  exponent 
of  b  in  the  divisor  is  greater  than  in  the  dividend,  and  we  have  1  —  2  for 
its  exponent  in  the  quotient.  This  gives  us  —  1,  and  we  have  the  quo- 
tient 2ab-]c-. 

139.  Here  we  find  the  sign  of  an  exponent  becomes  nega- 
tive, and  we  have  to  consider  what  meaning  we  are  to  attach 
to  the  signs  -f  and  —  when  appiied  to  exponents. 

In  this  example  the  exponent  became  negative  because  we  had  the 

factor  b  more  times  in  the  divisor  than  in  the  dividend,  and  we  could  not 

take  away  from  the  dividend  more  factors  of  that  kind  than  it  contained. 

We  might  have  taken  away  such  factors  as  were  in  the  dividend,  and 

expressed  the  division  we  could  not  perform.     This  would  have  given  us 

2«c2 
2ac-  -~  b,  or  — —  • 
0 

We   must   therefore  interpret  the   expression   2ab~1ci  as  meaning 

the    same ;    that  is,  the    negative    exponent    must    be    understood    to 

express  division  (Art.  54),  and  ab~2  will  equal  —  or  a  -*-  b-,  and 

b~2  =  1  x  6~2  =  —  =  1  j-  62.    Hence, 
Or 

140.  We  have  this  interpretation  of  the  force  of  the  signs 
+  and  —  affecting  an  exponent : 

A  positive  exponent  indicates  the  use  of  factors  as  multi- 
pliers ;  a  negative  exponent,  as  divisors. 

Note. — This  is  consistent  with  the  general  definition  of  these  signs, 
by  which  they  are  made  to  show  opposite  directions,  multiplication  and 
division  being  opposite  in  direction,  one  putting  in  and  the  other  taking 
out  a  factor.     (Arts.  81-91.) 

141.  By  the  analogy  between  the  coefficient  and  exponent, 
we  see : 

1st.  The  coefficient  shows  what  equal  terms,  and  the 
exponent  what  equal  factors,  are  used. 

2d.  The  sign  of  the  coefficient  shows  whether  the  equal 
terms,  and  the  sign  of  the  exponent  whether  the  equal  factors 
are  introduced  or  removed,  the  former  by  addition  or  subtrac- 
tion, the  latter  by  multiplication  or  division. 


44  D  I  V  I  S  I  0  X  . 

142.  It  must  be  understood,  also,  that  if  the  sign  —  be 
repeated  before  an  exponent,  it  again  reverses  the  direction  in 
which  the  factors  are  used. 

143.  We  may  have  what  is  equivalent  to  two  negative 
signs,  without  the  signs  themselves. 

Thus,  when  we  say  subtract  —  a,  the  word  subtract  is  equivalent  to 
the  sign  — ,  and  the  term  a  is  to  be  added  or  is  the  same  as  —'2a=  +  a. 

So  also  in  — ^,  the  ¥  is  a  multiplier,  its  position  below  the  line  being 

equivalent   to   the   sign   —    before  its  exponent,  which  with  the  sign 
already  there  gives  + .     It  might  therefore  be  written  ab*. 

144.  From  the  above  we  have 

at1  =  1  x  cr1  =  -•        That  is, 
a 

i°.  A  quantity  with  a  negative  exponent  is  equal  to  the 
reciprocal  of  the  same  quantify  with  a  like  positive  exponent. 

Again,  a  =  i  x  a  =  — •     That  is, 

2°.  A  quantity  with  a  positive  exponent  is  equal  to  the 
reciprocal  of  the  same  quantity  with  a  like  negative  exponent. 


Again,     a  x  j  =  a  •—  b,    and    a  ~  j  =  a '  x  b.     That  is, 

3°.  Multiplying  or  dividing  by  any  quantity  is  the  same 
as  dividing  or  mull! plying  by  its  reciprocal. 


Divide  the  following: 

2i//>f'-d 


6.     4a3bc2d~l  —  6a*b2cd. 

ga5b  VV 
7*      3<rV>r  H  >' 
8.      i$cPlfar*y  -^  sa'WxPy-1. 


DIVISION".  45 

145.  When  the  signs  of  the  dividend  and  divisor  are 
considered,  they  must  be  treated  as  the  other  factors,  in 
accordance  with  the  principles  already  established.  (Art.  128.) 
That  is, 

The  quotient  will  have  the  sign  — ,  with  an  exponent  equal 
to  its  exponent  in  the  dividend  minus  its  exponent  in  the 
divisor. 

Take  the  following  example : 

9.  Given    — ^ — 7-fj j^r) ; ;  ,    to    find    the 

—  2  («)  (—  b")  (—  c*)  (/r) 

quotient. 

Solution. — Cancelling  or  removing  the  factors  of  this  divisor  from 
the  dividend,  we  have  2d2b3di. 

In  the  same  way,  we  may  cancel  factors  of  direction,  remembering 
that  the  sign  +  ,  like  the  factor  i,  has  no  power. 

We  have  here  five  minus  signs  in  the  dividend  and  three  in  the 
divisor,  or,  what  is  the  same  thing,  — 5  and  — 3,  leaving  — 2,  or  +  for  the 
quotient.  It  will  he  readily  seen  that  the  same  result  would  be  given  by 
adding  the  exponents  of  — ,  for  whenever  the  difference  of  the  exponents 
is  odd,  the  sum  will  be  odd,  and  when  the  difference  is  even,  the  sum 
will  be  even,  hence  we  may  use  the  same  rule  for  the  sign  of  the  quo- 
tient in  division  as  for  the  product  in  multiplication.  (Art.  129.) 
Hence,  the 

Eule. — I.  An  even  number  of  negative  signs  gives  a  posi- 
tive quotient,  an  odd  number  a  negative. 

II.  If  there  be  but  two  signs  : 

Like  signs  give  +  ;  unlike  signs,  — . 

Note. — If  it  be  asked  why  these  signs  are  thus  treated  in  multipli- 
cation and  division,  the  answer  is,  that  such  use  of  them  meets  the  wants 
of  mathematical  analysis,  as  will  be  abundantly  illustrated  in  the  solution 
of  problems. 

Divide  the  following: 

10.  —  8ambn  -=-  2db. 

11.  6«5Z>4  -\ 3a b2. 

12.  —  $am+n  bm~n  -. anbm. 


46  DIVISION". 

13.  _8o  lb  2  -T-  2«-2^-4. 

14.  2^-2i-4  -. Sa^b~2. 

1,1  1,1 

15.  y^b*  -. tt*o*. 

16.  -a?{-ab)  +  $(<£$). 

17.  —  <r'//'  -=-  a  "  b  '  . 

jg_       am(m+n)})n(mTn)  _^_    —  (ab)mn. 
j  9.       _  0»is  §»"  _: am  ( —  &w)- 

20.  #2w  //~'71   —   .Tw_1  l/m+l. 

2 1 .  (afa)2^1  -f-  a"'1  bn  xn+\ 

CASE      II. 
146.    To  Divide  a  Polynomial  by  a  Monomial. 
A  polynomial  is  divided  by  a  monomial  by  the  following 

Ecle. — Divide  each  term  of  the  polynomial  by  the  divisor. 

Note. — Since  division  is  the  reverse  of  multiplication,  the  reason  for 
this  rule  will  appear  from  Art.  131. 

EXAMPLES. 
Find  the  value  of  the  following: 


(2aWx  —  4aW  +  60*8*2?)  -f-  2aW. 

(8anbm  +  4a2n63m  —  i2<74»&3w)  -j-  4anbm. 

(gambm  —  6anbm  +  I2a*mb2n)  -4-  3tf»4». 

(4rtTO-"  J*-'1  —  6a»+n  J»+ft)  -*-  2arnbn. 

Divide  the  same  by  2anb~n. 

Divide  the  same  also  by  2an~m  bn~s,  and  by  2«OTis. 

Divide  xv  —  yx  by  .?'//. 

Divide  a  m    +  Z>  "    by  ab. 

Divide  a T  +  b'  bv  a  />  . 


division.  47 


CASE       III. 

147.  To  Divide  a  Polynomial  by  a  Polynomial. 

i.  Divide  sub2  -f  yi2b  +  b'A  +  a5  by  a  +  b. 

Analysis. — ist.  Since  the  dividend  is  the  product  of  the  divisor  and 
quotient,  that  term  of  the  dividend  which  has  the  highest  power  of  a 
must  have  been  produced  by  the  multiplication  of  those  terms  in  the 
divisor  and  quotient  which  contain  the  highest  powers  of  a.  If,  there- 
fore, we  divide  that  term,  of  the  dividend  containing  the  highest  power 
of  a  by  the  corresponding- term  of  the  divisor,  we  shall  find  one  term  of 
the  quotient. 

2d.  Multiplying  the  divisor  by  this  term  of  the  quotient  and  sub- 
tracting the  product  from  the  dividend,  we  shall  have  a  remainder  to  be 
divided  as  before. 

3d.  Whenever  this  remainder  becomes  zero,  the  division  will  be 
complete. 

4th.  The  division  may  be  stopped  at  any  time  and  the  quotient  be 
completed  by  adding  the  remainder  over  the  divisor  in  the  form  of  a 
fraction  to  indicate  the  uncompleted  division. 

It  will  be  more  convenient  in  dividing  to  arrange  the  dividend  and 
divisor  in  the  order  of  the  ascending  or  descending  powers  of  the  same 
letter. 

OPERATION. 

Dividend,  a3  +  ^aPb  +  ytb-  +  b3  |  a  +  b    Divisor. 

ist  product,         a3  +    (Pb a2  +  2ah  _,.  52    Quotient, 

2a26  +  306s  +  b3 ist  remainder. 

2d  product,  2a-b  +  2«62 

+    ab'2  +  b3 2d  remainder. 

3d  product,  +    ob1*  -f-  b3 

148.  \Ye  have  from  the  above  illustration  the  following 

Rule. — I.  Arrange  the  terms  of  both  dividend  and  divisor 
in  accordance  ivith  the  ascending  or  descending  powers  of  the 
same  letter. 

II.  Divide  the  first  term  of  the  dividend  by  the  first  term  of 
the  divisor,  and  write  the  result  for  the  first  term  of  the 
quotient. 

III.  Multiply  the  divisor  by  this  term  of  the  quotient,  and 
subtract  the  product  from  the  dividend. 


48  DIVISION". 

IV.  Divide  this  remainder  as  before,  and  so  on  till  the 
division  is  complete  or  a  remainder  is  found  which  has  no  term 
di  risible  by  lite  first  term  of  the  divisor. 

V.  Write  t lie  final  remainder,  if  any,  over  the  divisor  in  the 
form  of  a  fraction  and  add  it  to  the  quotient. 


EXAMPLES. 

2.  Divide     a5  —.satb  +  loaW  —  io«2£3  +  ydS  —  b5    by 
a?  —  2ab  +  b\ 

OPERATION. 

a5— 5^45  +  io«3&5—  ioa263+5<?7>4— £>5    a-  —  inb  +  b-  Divisor. 

a5-2afib+     aW a3-3«-&  +  3a#-63,     Quot. 

ist  rem.,        — 3<z46+   ga3fr2  —  ioa263 
—  3«4&+   6a36-—   3rt-i3 

2d  rem.,  3«362—   jiM3  +  $«& 

3a3&2-  6«%3  +  3^4 


3d  rem.,  —     rf-63  +  2^4  — 65 

—     a263  +  2a64— 65 

3.  Divide     £6  —  2X5  +  2o.r3  —  33a;2  —  514  -}    142  —  3     by 
3  —  2X  +  a:2. 

4.  Divide  a:8  —  ?/8  by  :r2  —  y2. 

5.  Divide  x6  -f  a6  by  se2  +  «2. 

6.  Divide  «  —  x  by  am  —  x*. 

D.     .  -.  r.  3  s  1  -    .  .i  1  1  3 

lvide  a-  —  a?x  +  as* — 2<r-\r~  +  x~  by  «-• —  ax*  +  «%— #t. 

8.  Divide  x*n—a2n  by  .<•"  —  «n. 

9.  Divide  am ' »  &»  —  4a"1 '  »-i  Z»2«  —  30t»h»-2£3»  _j_  (5am+«-3£4« 
by  am  —  zaP1'1  bn  —  6am~*  b2n. 

10.  Divide  rr1  +  lrl  by  a-^  +  b~k 

11.  Divide  <75  —  50*3  +  2^-V2  —  sa&c8  +  ax4  by  «2  +  z2. 

12.  Divide  .t2//  —  $xy  —  x  -f  2xy2  by  .17/  —  3//  —  1  -f  2//2. 


DIVISION.  49 

13.  Divide  1  —  32;  -f  3-r-  —  a;3  by  1  —  x. 

14.  Divide  b3  +  $bx2  +  $tPx  +  .r3  by  b  +  x. 

15.  Divide  x4  +  syi  —  6a?y  by  x  —  3!/- 

16.  Divide  6ft4  +  4«  —  10ft3  —  15  by  3^  —  2a  +  1. 

17.  Divide  2ft4  +  iift3x'  +  2oa2x2  +  i3ft.z3  +  2^ 

by  ft2  +  3«.r  +  2  a;2. 

18.  Divide  ft4  —  4ft3&  +  6aW  —  \ab*  +  ¥  by  a*—2ab  +  Vi. 


ABBREVIATED  OPERATION. 

a2 

a4  —  4«36  +  6<2262  —  4ab3  +  64 

+  2«5 

+  2«3&  —  4a26a  +  2rt63 

-&* 

—    a2Z>2  +  2a¥  —  b* 

—  2a3b  +    «262  +0     +0 

Quotient,        a-  —  2ab   +  62 

Analysis. — 1st.  Divide  a4  by  a2  and  set  down  the  quotient  a2. 

2d.  Multiply  the  several  terms  of  the  divisor,  except  the  first,  by  this 
quotient,  and  set  the  products  2azb  and  —  «2&2  as  above. 

3d.  Add  the  second  column  and  set  the  sum  —2a3b  below. 

4th.  Divide  this  by  the  first  term  of  the  divisor,  and  proceed  as 
before. 

149.  By  this  method,  the  dividend  and  divisor  must  be 
arranged  in  accordance  with  the  ascending  or  descending 
powers  of  each  letter  ;  that  is,  tbe  exponent  of  each  letter  must 
either  increase  or  decrease  by  unity  from  left  to  right  in  both 
dividend  and  divisor. 

Terms  may  be  inserted  with  o  for  coefficients,  if  by  so  doing  the 
exponents  may  be  made  to  form  an  increasing  or  decreasing  series. 

It  will  also  be  observed  that  the  signs  of  the  divisor,  except  the  first, 
have  been  changed,  which  enables  us  to  add  the  products  instead  of 
subtracting. 

The  product  obtained  by  multiplying  the  first  term  of  the  divisor  is 
not  written,  since  it  always  cancels  a  term  of  the  dividend. 

The  other  products  are  written  each  under  a  similar  term  of  the 
dividend,  and  in  line  with  that  term  of  the  divisor  from  which  it  was 
obtained. 

Let  the  student  work  in  this  way  such   examples  as  are 
suited  to  this  process,  found  in  Art.  148. 
3 


50  DIVISION 


DIVISION     BY     DETACHED     COEFFICIENTS. 

150.  The  .work  of  division  may  often  be  abbreviated  by 
dropping  the  literal  factors  and  replacing  them  in  the  cpuotient. 
This  can  only  be  done  when  the  dividend  and  divisor  can  be 
arranged  according  to  the  ascending  or  descending  powers  of 
each  letter,  as  in  multiplication.     (Art.  132.) 

1.  Divide  a4  —  b4  by  a  —  b. 

Tliis  may  be  written 

a4  +  oayo  +  oa2&2  +  oab3  —  ¥, 

in  which  the  exponents  of  a  decrease  and  those  of  b  increase  by  unity  in 
the  successive  terms.  Writing  the  coefficients  only,  and  dividing,  we 
have, 

1+0  +  0  +  0—  1   '   1  —  1         Coef.  of  Divisor. 

1  —  1 


1  +  1  +  1  +  1     Coef.  of  Quot. 


+  1+0 
1  —  1 


1  +  o 
1  —  1 


1  —  1 
1  —  1 

It  is  evident  by  dividing  a4  by  a  that  the  first  term  of  the  quotient 
will  be  a3,  and  we  may  then  write 

n3  +  a"b  +  ab2  +  ft3,    Ans. 

EXAMPLES. 

Divide  by  detached  coefficients : 

2«3  —  6a2x  4-  6ax2  —  2X3  by  2a  —  2X. 


cfi  —  b3  by  a2  +  ab  +  &■ 

aA  —  x*  4-  20.2?  —  (fix*  by  ax  +  a?  —  x2. 

x4  —  4-r3  +  Gx2  —  42  +  i  by  x  —  1 . 

x5  +  x4!/  —  5>'3#2  +  6-ry4  +  2  if  by  x2  4-  ^xy  +  y2. 

1  —  4b  4-  ioZr  —  16b3  +  17&4  —  12b5  by   1  —  2b  +  3J2. 

2uk  +  wdhc  +  20((2x2  4-  ^rtrc3  +  2.C4  by  a2  +  3«.r  4- 2Z2. 


DIVISION.  51 


SYNTHETIC      DIVISION. 

151.  The  operation  may  be  still  further  shortened,  when 
the  first  coefficient  of  the  divisor  is  unity,  by  using  detached 
coefficients,  in  the  manner  of  Art.  148,  Ex.  18. 

5.  Divide  «4  —  4a8*  +  6aW  —  ^a¥  +  ¥  by  «2  —  20b  +  b\ 

OPERATION. 


I 

+   2 
—   I 


i—  4  +  6  —  4  +  1 
+  2  —  4  +  2 

—  1  +  2  —  1 


1  —  2  +   1 
.•.    Quotient  =  a?  —  iab  +  b2. 
Note. — Observe  that  the  coefficient  of  each  remainder  becomes  the 
coefficient  of  a  term  of  the  quotient.     This  method  is  called  Synthetic 
Division* 

In  like  manner  divide  the  following: 

6.  a6  +  2a3b3  +  b6  by  «2  —  ab  +  P. 

7.  x3  +  x2y  —  xy2  —  ys  by  x  —  y. 

8.  a5  -+-  5«4£  +  \oa%x2  +  locfcfl+saxl+a?  by  a2-\-2ax  +  x2. 

The  method  of  Synthetic  Division  is  especially  convenient 
in  case  of  a  binomial  divisor.     Thus, 

9.  Divide  x5  —  2X*  +  53?  +  3a;2  —  122;  —  28  by  x  —  2. 

OPERATION. 
I 
+   2 


I 

— 

2 

+ 

5 

+ 

3 

— 

12 

— 

28 

+ 

2 

+ 

0 

+ 

10 

+ 

26 

+ 

28 

I 

+ 

O 

+ 

5 

+ 

*3 

+ 

14 

.*.    a?1 +  00?  + 5a:2 +  132+ 14  =  a?1  +  5a;2 +  132+ 14  =  Quotient. 

Or,  transferring  the  divisor  to  the  right  of  dividend,  and  omitting  the 
first  term,  an  arrangement  frequently  more  convenient,  we  shall  have 

1  —  2  +  5+     3  —  12  —  28  I  2 

+  2  +  0+10  +  26  +  28 
1+0  +  5  +  13  +  14 
And        x*  +  5X2  +  132:  +  14  =  Quotient. 

*  It  was  first  proposed  by  W.  G.  Horner,  of  England.    (See  p.  307,  Note  6.) 


52  DIVISION. 

10.  Divide  x5  —  32  by  x  —  2. 

I+O  +   O  +  O+      O  —  32  I  2 
+   2   +   4   +   8   +    16   +   32 


1  +  2  +  4  +  8  +  16 
x1  +  2a;3  +  4a:2  +  8x  +  16,    Ans. 

11.  Divide  a4  —  2a;2  +  32;  —  7  by  x  +  3. 

1+0-2+    3-    7  I  -  3 
-3  +  9-21  +  54 


1  -  3  +  7  -  18  +  47 


cr3  —  3.C2  +  7^  —  18  h — — ,    Ans. 

x  +  3 

Note. — In  this  case  we  have  the  remainder  47,  and  therefore  the 
division  is  incomplete. 

Divide  in  like  manner: 

1 2.  a;6  —  3a4  +  2cc2  by  2  —  1. 

13.  x5  +  jx4  —  3Z2  +  -jx  —  5  by  x  —  3. 

14.  a4  —  2a;3  +  82;  —  16  by  a*  —  2  ;    also  by  k  +  2. 

15.  x5  —  5-r3  -f-  2a;2  —  7  by  a;  +  1. 

16.  x7  —  5a;5  +  2x2  —  1  by  x  +  3. 

17.  z6  —  4a;5  +  Ox4  —  Ox2  +  42;  —  1  by  z— 1  and  by  x+  1. 

18.  .r5  —  7a;3  +  6x  —  5  by  x  +  2  and  by  a;  —  3. 

19.  .r3  —  1   by  x  —  1  and  by  x  +  2. 

20.  .r7  —  1  by  a:  —  1  and  by  x  +  1. 

21.  an  +  a;12  by  a*  +  a4  and  by  a3  +  a^. 

22.  a40  +  yw  by  JB2  +  if  and  by  a'5  +  if. 

23.  a-14  +  1  by  x2  +  1  and  by  x"  +  1. 

24.  xw  +  a36  by  a:6  +  a«  and  by  a;2  +  «4. 

25.  2?  +  y«  by  a-3  +  ?/' ;    also,  «5  +  bw  by  a  +  62. 

26.  a-5  —  6x*  +  5a-3  —  7a:2  —  4X  +  9  by  x—  3  and  a- -+3. 
2 7-  -''7  +  3''3  —  2a4  —  5_/'3  +  33  +  r  by  a;  +  1  and  a-—  r. 
28.  a;6  +  4.f5— 7.i4  +  2ar3— 7a:2  +  4.t+i6  by  x+i  and  x—  1. 


CHAPTER    V. 

FACTORING. 

152.  Factoring  is  the  process  of  separating  a  quantity 
into  factors. 

Unity  having  no  power  as  a  factor,  will  not  be  considered 
as  a  factor  in  the  following  discussions. 

A  Prime  Factor  is  one  which  does  not  contain  other 
factors. 

Factors  are  prime  to  each  other  when  they  have  no 
common  factor. 

153.  The  operation  of  factoring  is  performed  either  by 
inspection,  in  which  we  employ  our  previous  knowledge  of  the 
forms  of  products ;  or  by  trial,  in  which  we  determine  by 
division  whether  one  quantity  is  a  factor  of  another,  and  at 
the  same  time,  discover  the  other  factor. 

Note. — Finding  the  equal  factors  of  a  quantity  is  the  work  of 
Evolution,  which  will  be  treated  in  its  place.  We  are  at  present  con- 
cerned especially  with  unequal  factors,  though  equal  factors  may  often 
be  found  by  the  same  methods. 

154.  The  Factors  of  a  Monomial  are  too  easily 
discovered  to  need  explanation,  since  they  are  all  indicated  by 
the  exponents.  The  same  may  be  said  of  the  monomial  factors 
of  a  polynomial,  since  they  must  be  factors  of  each  term  of  the 
polynomial. 

Thus,  a"o  is  a  factor  of  iazo  +  yiW  —  5«363,  and  we  may  write 
a"b(2a  +  3&2  —  5a62). 

155.  The  Factoring  of  Binomials  will  be  facili- 
tated by  the  following  theorems : 

Theorem  I. — The  product  of  the  sum  and  difference  of  any 
two  quantities  is  equal  to  the  difference  of  their  squares. 
Demonstration,    (a  +  b)(a  —  b)  =  ai  —  b\ 


54  FACTORING. 

Theorem  II. — The  difference  of  any  two  quantities  is  a 
factor  of  the  difference  of  any  like  positive  integral  powers  of 
the  same  quantities. 

Demonstration. — Let  a  and  6  represent  the  quantities,  and  a11  and 
bn  the  like  powers,  in  which  n  is  a  positive  integer.  We  are  to  prove 
that  a'1  —  bn  is  divisible  by  a  —  b. 

Dividing,  «"  ~  b"  |  a  —  b 

an  _  ad-i  i  ^n_l 

We  have  the  remainder,  an~l  b  —  b" 

One  of  the  factors  of  the  remainder  is  the  difference  of  like  powers  of 
the  same  quantities,  whose  exponent  is  one  less  than  in  the  dividend. 
If  this  factor  be  divisible  by  a  —  b,  the  dividend  is  also  divisible  by  it. 

We  have  therefore  proved  that  if  the  Theorem  be  true  for  one 
value  of  n,  it  is  also  true  for  a  value  one  unit  greater. 

For  example,  if  it  be  true  when  71=2,  it  will  also  be  true  when  11=3, 
and  for  the  same  reason  when  n  =  4,  5,  6,  etc. 

But  we  know  that  a  —  b  is  a  factor  of  a-  —  b'2 ;  that  is,  the  theorem 
is  true  when  n  =  2.     Hence,  it  is  universally  true. 

Cor.  i. — The  sum  of  two  quantities  is  a  factor  of  the  differ- 
ence of  any  like  even  positive  integral  powers  of  the  same. 

That  is,  a-n  —  62"  is  divisible  by  a  +  b.     For, 
a2"  -  bin  =  (a2)"  -  {¥)", 
which  by  the  Theorem  is  divisible  by 

a-  —  b'2  =  (a  +  b)  (a  —  b). 

Cor.  2. — TJie  difference  of  any  tivo  powers  whose  exponents 
have  a  common  factor,  may  be  separated  into  factors  one  of 
which  is  a  binomial. 

For,  amn  —  &'"■  =  (a"')n  —  (bT)n,  which  has  am  —  bT  as  a  factor. 

Theorem  III. —  The  sum  of  any  two  quantities  is  a  factor 
of  the  sum  of  any  like  odd  positive  integral  powers  of  the  same. 

That  is,  a2"  H  +  b-n+l  is  divisible  by  a  +  b.     For,  dividing, 
«"" M  +  I  I  a  +  b 

We  have  the  remainder,  —  a'2"b  +  b-"+l  =  (a2n  —  b-")  (—  b). 

But    a2"  —  b'2n    has  been   Bhown    to   be  divisible  by   a  4-6;   hence, 
frtn+i  ja  aiso  divisible  by  a  +  b. 


FACTOEING.  55 

Cor. — Tlie  sum  of  any  two  powers  each  of  whose  exponents 
has  the  same  odd  factor,  may  be  separated  into  factors  one 
of  which  is  a  binomial. 

That  is,  a^n+1'>m  +  &0+1)--  has  a  binomial  factor.  For  this  may  be 
written,  (a'")2n+I  +  (brf+\  which  has  the  factor  a"'  +  br. 

For  example,  a6  +  b*  =  (a-)s  +  (b3f  =  (a'2 +  b3)(ai-aW  +  a6). 

It  is  evident  that  m  and  r  may  be  equal ;  as,  a6  +  b''  —  (a2)3  +  (b'2)3, 
which  has  the  factor  a2  +  ¥. 

They  may  also  be  negative  ;  as,  a~b  —  b~10  =  (a-1)5  —  (&-2)5»  which 
has  the  factor  a-1  —  6~2. 

Factor  the  folio  wins: : 


I. 

ai  -  V. 

2. 

a3  —  b\ 

3- 

a3  +  b3. 

4- 

a~*  -  b~K 

5- 

fl-3  _  J-3. 

6. 

a~3  —  b\ 

7- 

a6  —  b~6. 

8. 

a6  +  ¥. 

9- 

a12  +  b15. 

IO. 

a12  -  bvi. 

1 1. 

x6  +  #9. 

12. 

a;-6  —  y~9. 

13- 

xG  +  y~9. 

^ws.  (a2  +  V)(a  —  b)  (a  +  b). 

Ans. 

(a  -  J)  (a2  +  ad  +  £2). 

Ans. 

(a  +  J)  (a2  -  ab  +  J3). 

14. 

z10  +  #15. 

T5- 

a;12  +  I/21. 

16. 

or12  —  ?/9. 

17. 

2-14    _|_    yU 

18. 

XU  -   I. 

19. 

I    4/    ff10. 

•  20. 

I   —  a"10. 

21. 

a72  —  £96. 

22. 

jplOO   +    yM 

156.  To  find  the  binomial  or  polynomial  factors  of  a 
polynomial,  the  forms  of  products  must  be  observed.  The 
following  theorem  will  aid  in  this  inspection  : 

Theorem  IV.  —  The  square  of  a  polynomial  is  equal  to  the 
sum  of  the  squares  of  its  terms  plus  ttvice  the  sum  of  the 
products  of  the  terms  taken  two  and  two. 

Demonstration-,  (a  +  b  +  c)9  =  a'2  +  20b  +  lac  +  b'2  +  2bc  +  c2,  as 
is  shown  by  performing  the  multiplication,  and  the  process  is  such  as  to 
make  it  evident  that  the  same  would  apply  to  a  polynomial  of  any  num- 
ber of  terms.     Hence, 


56  FACTORING. 

Cor.  i. —  Tlte  square  of  (he  sum  of  two  quantities  equals  the 
sum  of  their  squares  plus  twice  their  product ;  and  the  square 
of  the  difference  equals  the  sum  of  their  squares  minus  twice 
their  product.     Thai  is, 

(a  +  bf  =  a?  +  2ab  +  J2,     and 

(a  —  b)2  =  dl  —  2ab  +  b2. 

Cor.  2. — If  two  polynomials  having  like  terms,  but  ruith 
unlike  signs  be  multiplied  together,  the  product  icill  be  the 
same  as  the  square  of  one  of  the  polynomials  ;  except, 

ist.  The  squares  of  the  terms  having  unlike  signs  in  the 
two  polynomials  will  be  negative. 

2d.  The  double  products  formed  by  those  terms  which  have 
like  signs  in  one  polynomial  and  unlike  in  the  other  will  not 
be  found  in  the  result ;  and  the  double  products  formed  by 
terms  having  like  signs  in  each  polynomial,  but  unlike  in  the 
two,  will  be  negative. 

157.  From  this  theorem  and  its  corollaries  factor  the 
following  examples: 

23.  4a2  —  gb2  +  c2  +  4ac. 

Analysis. — The  terms  which  are  perfect  squares  suggest  the  terms 
of  the  factors  sought.  These  give  2a,  36,  and  c.  The  only  double 
product  found  is  40c  ;  hence,  2a  and  c  have  either  like  or  unlike  signs  in 
both  factors,  but  2a  and  3?;,  and  3ft  and  c  have  unlike  signs  in  one  and 
like  signs  in  the  other  factor.  This  requires  for  the  factors,  2a  —  36  +  c 
and  2a  +  36  +  C 

24.  4a2  —  gb2  +  6bc  —  c2. 

25.  4«2  —  4«c  —  gb2  +  c2. 

26.  a2  +  2ab  +  b2  +  ac  +  be. 

Observe  that  the  first  three  terms  are  the  square  of  a  +  b,  which  is 
also  a  factor  of  the  last  two  terms. 

27.  4  +  46  +  b2  +  2f  +  be. 

28.  a2  4-  Ga  4-  8. 

Separate  into  terms  thus  :  as  +  4^  -1-  4  4-  2a  +  4. 


FACTORING.  57 

29.  (fix*  +  2ttX?  +  a4  —  a4  4-  2(fix  —  x%. 

30.  4a2  —  4a, v  —  24a;2. 

3 1 .  qcfix2 — 4CIX3 — 4a4  4-  402%  4-  ^xhf — ahf  —  2ay3— y*  4-  a4. 

32.  «6  —  ay  4-  5a  4-  as  —  ay  4-  ?/2  4-  2:2:  —  by  —  yz. 
33-  «2  +  3«^  +  2^2- 

158.  By  multiplying  (x  +_ax)  («  +  a2) (»  +  O, 

and  collecting  the  terms  containing  the  like  powers  of  x,  we 
shall  find, 

1st.  The  highest  power  of  x  will  be  xn,  and  its  coefficient 
will  be  1. 

2d.   The  coefficient  of  xn~ 1  will  be  al  4-  a2  4-  «3  .  .  .  .  «n. 

3d.  The  coefficient  of  a;0  will  be  ax  a2  .  . . .  #». 

Let  the  student  illustrate  this  by  multiplying  five  or  six 
such  factors. 

159.  From  this  we  may  often  discover  the  binomial 
factors  of  a  polynomial  function  of  a  single  letter. 

34.  Required  to  factor  x2  —  2X  —  15. 

Solution. — The  factors  of  this  will  have  numerical  terms  whose 
product  is  —  15,  and  whose  sum  is  —  2,  and  we  find  that  3  and  —  5 
fulfill  these  conditions.     The  factors,  therefore,  are  x  +  3  and  x  —  5. 

35.  Factor  yfi  —  ga~  —  12a  4-  36. 

Solution.— Dividing  hy  3  to  make  the  coefficient  of  a3  unity,  we 
have,  of'  —  3a2  —  4(1  +  12.  The  product  of  the  numerical  terms  of  the 
factors  is  12,  their  sum  is  —  3,  and  there  are  three  of  these  factors.  The 
only  three  factors  whose  product  is  +  12  and  sum  —  3  are  +  2,-2,  and 
—  3.  These  give  a  +  2,  a  —  2,  and  a  —  3,  which  with  the  3  already 
taken  out,  are  the  factors  sought. 

Find  the  factors  of  the  following : 


36. 

a:2  —  2X  —  35.                       39.     4.? 

•2  +  32X  4-  60. 

37- 

x*  4-  x  —  20.                      40.     5a 

■2  4-  152  —  140. 

38. 

x2  —  gx  4-  20.                    41.     xz 

—  x2  —  142;  4-  24« 

42. 

x3  —  8a;2  4-  112:  4-  20. 

43- 

5^  —  35*3  +  85.1'2  —  853  4-  30. 

44. 

2ttX3  4-   2«.t'2  —   7,2aX  4-   40a. 

45- 

x5  4-  a4  —  5xz  —  5a-2  4-  4X  +  4. 

CHAPTER    VI. 

DIVISORS    AND     MULTIPLES. 

160.  The  Factors  of  a  quantity  are  sometimes  called 
its  Divisors,  since  it  may  be  divided  by  any  one  of  them. 

161.  Commensurable  Quantities  have  a  common 

divisor;  as,  ac  and  ax,  both  of  which  have  the  divisor  or 
facto?'  a. 

162.  Incommensurable  Quantities  have  no  com- 
mon divisor ;  as,  abc  and  :ryz.  Such  quantities  cannot  be 
measured  with  the  same  unit ;  hence  the  name.     (Art.  8.) 

GREATEST     COMMON     DIVISOR. 

163.  The   Greatest  Common   Divisor  of  two  or 

more  quantities  is  that  common  divisor  which  contains  the 
greatest  number  of  factors. 

164.  If  the  prime  factors  of  the  quantities  can  be  discovered 
by  inspection,  the  f/,c.d.  may  be  found  by  combining  all  the 
common  factors  into  a  product,  using  each  as  many  times  as  it 
is  found  in  every  one  of  the  quantities. 

165.  When  these  factors  cannot  be  thus  discovered,  we 
may  employ  a  process  based  on  the  following 

Theorem.—  A  factor  common  to  two  quantities  is  a  factor 
of  tlw  remainder  resulting  from  the  division  of  one  of  the 
quantities  by  tin'  <>//<rr. 

Pkmonsthation.  -Let  a  and  h  represent  the  quantities,  and  let  q 
and  r  be  the  quotient  and  remainder  obtained  by  dividing  a  by  b.  Then 
will  a  —  bq  +  r  or  a—bq  =  r.  Now  every  factor  common  to  a  and  b  is  a 
factor  of  a—bq,  and  therefore  of  its  equal  r. 


DIVISORS.  59 

Cor. — The  g,  c.  d.   of  b  and   r  will  be  the  g.  v,  d.  of 

a  and  b. 

For,  since  a  —  bq  +  r,  every  divisor  of  a  and  b  will  be  a  divisor  of  b 
and  r,  and  every  divisor  of  b  and  /•  will  divide  a  and  &.     Hence, 

166.  To  find  the  g.  c.  d.  of  two  quantities  we  have  this 

Rule. — I.  Find  by  inspection  as  many  common  factors  of 
the  two  quantities  as  possible,  and  removing  them  from  the 
quantities,  reserve  them  as  factors  of  the  g.c.  d. 

II.  Reject  also  from  each  quantity  all  prime  factors  not 
common  to  both. 

III.  Divide  one  of  the  quantities  remaining  after  these 
factors  are  removed  by  the  other,  and  if  there  be  a  remainder, 
divide  the  divisor  by  the  remainder,  and  the  second  divisor  by 
the  second  remainder,  and  so  on,  until  a  divisor  be  found  ivhich 
gives  no  remainder.  The  product  of  this  divisor  with  the 
co?mnon  factors  already  removed  loill  give  the  greatest  common 
divisor. 

Notes. — i.  If,  in  the  course  of  the  operation,  common  factors  are 
discovered,  they  should  be  removed,  and  reserved  for  the  (/.  c.  d. 

2.  If  prime  factors  not  common  are  discovered,  either  in  the  dividend 
or  divisor,  they  should  be  rejected  before  dividing. 

167.  If  any  division  gives  a  fractional  term  in  the  quotient, 
thus  involving  fractional  products,  this  may  be  avoided  by 
multiplying  that  dividend  by  such  a  number  as  will  make  its 
first  term  divisible  by  the  first  term  of  the  divisor.  This  will 
not  affect  the  g.  c.  d.  unless  it  introduce  a  new  common 
factor.  This  it  will  not  do,  if  the  second  part  of  the  rule  has 
been  followed. 

Note. — The  f/.r.  d.  will  have  the  double  sign  (±);  for,  if  +  a  be  a 
divisor  of  a  quantity,  —a  will  also  be  a  divisor.  Hence,  positive  or  nega- 
tive factors  may  be  rejected  or  introduced. 

168.  The  Greatest  Common  Divisor  of  Polynomials. 

The  greatest  common  divisor  of  polynomials  may  be  found 
by  the  preceding  rule,  as  illustrated  in  the  following  examples : 


60 


DIVISORS, 


Find  the  {/.  r.  (I.  of  the  following  polynomials: 
i.     1225  —  5i.r3  -4-  i2.r  and  2.?5  —  4X%  —  2.r3  +  4a;2. 

Analysis.  33*  is  evidently  a  factor  of  the  first,  and  2.r-  of  the  second. 
Thc3  factor  x  being  common  to  both,  must  be  reserved  as  a  factor  of  the 
greatest  common  denominator,  but  3  and  2X  may  be  rejected.  We  there- 
fore divide  the  first  by  3c  and  the  second  by  23'-,  and  proceed  in  accord- 
ance with  the  rule,  as  follows  : 


42^+0 

4X4  —  8x3  —    43-'-  +  8.c 


OPERATION. 

17.!'-    +    O     +      4      SB8  —  2.T2  —  X   +   2 

4X  +  8 


Sx3  —  13.T-  —  S.r  +     4 

83>3  —  i6r2  -  8.r  +  16 

+     32:'-'  —  12 

Rejecting  the  factor  3  from  the  remainder,  and  dividing  divisor  by 
remainder, 

x3  —  2X2  —     x  +  2     x-  —  4 

^_ 4X  x  —  2 

—  2X-    +   2>X   +    2 

-  23?  +8 

33-  —  6 
Rejecting  the  factor  3,  we  have  the  remainder  x  —  2,  which  will  of 
course  divide  x-  —  4.     (Art.  155.)     Therefore  the  f/.  c.  d.  is 

±  (x  —  2)  x,      or       ±  (a?  —  2x). 

2.     4-f3  —  6.r'2  —  4X  +  3  and  2.C3  -j-  .r2  -j-  x —  1. 
The  operation  may  be  put  in  the  following  convenient  form. 


OPERATION. 


1st  dividend,  4X3  —  6x'2  —  47-  +  3 
4.r3  +  2x-  +  2x  —  2 
2d  divisor, 
2d  quotient, 
3d  dividend, 


4th  divisor, 


-  8 

r- 

-  6a 

4 

5 

-  S.j:2 

-  8a;2 

+ 

6x 
36a; 

+ 

X 

5 
16 

-  21  ) 

- 

42.r 

+ 

21 

2a; 

- 

1 

±  (2X 


—  X  +     4 

1)  =  q.  c,  (I.     A  us. 


2Xi   +    X2   +   X 


1st  divisor. 

2  1st  quotient. 

8.T3  +  4a;2  +  4a;  —  4  2d  dividend. 
Sx3  +  6x*  —  53; 

—  2.r2  +  gx  —  4  3d  divisor. 

4  3d  quotient. 

—  2.C2  +  9a1  —  4  4th  dividend. 

—  23!2  +     x 


+  8x  —  4 
8.r  -  4 


MULTIPLES.  Gl 

3.  ««4  +  2a2«3  +  a3^2 — ax2 — 2«2a; — a3  and  a6 — 2a4a:24-a2ar*. 

4.  x4  4-  io.c3  4-  24a;2  —  10a;  —  25  and  4a;4  —  212:3  +  5. 

5.  4«2^3  +  a3b  +  2aW  +  2a¥  +  «2&2  —  b4 

and  a3Z>  —  aW  —  a&3  +  6*. 

6.  (a4  —  £4)  aa;  and  (a3  +  63)  5a;. 

7.  (a3  —  Z»3)  (a  —  x)  and  (a2  —  J2)  (a  4-  a;). 

8.  3a;5  +  7a;3  —  5a:2  4-  3  and  3.T3  —  2a;2  —  1. 

9.  o2  —  $az  +  4a;2  and  a3  —  a2x  4-  3«a;2  —  32s. 

10.  x4  —  6x3  4-  13X2  —  \2x  4-  4  and  a^  —  4a;2  4-  52;  —  2. 

169.  The  g.  c.  d.  of  more  than  two  quantities  may  be 
found  by  finding  the  g.  c.  d.  of  two,  and  then  of  this  g.  c.  d, 
and  a  third,  and  so  on  till  all  are  used. 

Find  the  g.  c.  d,  of  the  following : 

11.  x2  —  x  —  6  ;   x2  4-  4X  4-  4 ;   a;2  —  4. 

12.  3.T24-6a;  +  3;  6a;2 — 30a; — 36;  g:i?-{-2'jz-\- 18;  12a;2 — 12. 

13.  a;3  4-  4a:2  4-  6x  4-  9 ;  x3  4-  x2  —  2a;  4-  1 2 ;  x2  —  % — 12. 

14.  x4 — x3 — 4X24-i6a; — 24;    x3 — 5a;2 4- 8x — 4;   x2 — 2a; — 8.. 

15.  xi  —  8a;2  4-  16  ;  x3  4-  2a;2  —  4a;  —  8 ;  x3  —  2X2  —4X 4- 8. 


LEAST     COMMON     MULTIPLE. 

170.  A  Multiple  of  a  quantity  is  the  prochict  of  that 
quantity  by  any  factor.    Hence, 

It  is  any  quantity  which  is  divisible  by  the  given  quantity. 

A  Common  Multiple  is  a  multiple  of  several  quantities. 

The  Least  Common  Multiple  is  the  quantity  which 
contains  no  factors  except  those  which  are  necessary  to  make 
it  a  multiple  of  the  several  quantities. 

The  I.  c.  m.  will  therefore  contain  every  factor  found  in 
the  given  quantities,  and  each  factor  will  be  found  in  the 
I.  c.  in.  as  many  times  as  it  is  found  in  any  one  of  the  given 
quantities.     Hence,  • 


62  MULTIPLES. 

171.  To  find  the  I.e.  in.  of  several  quantities  we  have 
this 

Eule. — I.  Separate  the  quantities  into  their  prime  factors. 

II.  Give  each  of  these  factors   an  exponent   equal   to  the 
largest  exponent  it  has  in  any  of  the  given  quantities. 

III.  The  product  <f  the  quantities  thus  obtained  will  be  the 
least  common  multiple  sought. 

172.  When  the  least  common  multiple  of  hvo  quantities 
is  required,  it  is  evidently  equal  to 

Hie  product  of  the  quantities  divided  by  their  g.  c.  d.     Or, 

One  of  the  quantities  divided  by  their  g.c.d.  and  multi- 
plied by  the  other.  N 

For  in  the  product  of  two  quantities  the  ff.  C.  (I.  will  be  found  twice, 
and  it  is  only  necessary  that  it  should  be  found  once. 

173.  Find  the  I.  c.  m.  of  the  following: 

i .  x?  —  a3;    x2  —  a2 ;    x  +  a  ;    x  —  a. 

2.  $a2x3y ;    6asx2y2;    2axy3. 

3.  ,"2  4-  2CVX  4-  a2;    a  +x ;    a  —  x. 

4.  x2  —  1 ;    x  4-  1 ;    x  —  1 ;    x2  4-  ?.x  -\-  1. 

5.  x3  —  1 ;    x2  -\-  x  —  2  ;    x  —  1. 

(3#       zni/m  '     tfm+n  i/m+n  •     xmVn. 

7.  (rf-i)»j    (x-1)2;    z+  1. 

8.  a3  +  .r3 ;    a2  —  x2 ;    a2  —  ax  4-  x2. 

9.  a3  —  .r3 ;    a2  —  x2 ;    a2  -f  i7.r  4-  a^. 

10.  rt2  —  .r2;    a2  —  2ax  —  .r2;    a2  +  2(tx  +  a-2. 

11.  ai  —  1  ;    a3  +  a2  +  a  -\-  1  ;    a3  —  a2  +  a  —  1  ;  a2  4-  1 . 

I2.  JC6    _   y6.        _?4    _|_     r2//2    _|_    y   .        38    +    yS.        x2    +    y2 

1 3.  a-6  +  «6 ;    .r4  —  «4 ;    .r2  4-  a2 ;    a;  +  a. 

14.  .r3  4-  a^  —  1 02;  4-  S  ;    x2  +  2x— 8  ;    a-2— 3.1- 4-  2  ;    .r2—  1 . 

J5-     -''4  +  5-1'3  +  5-?'2  —  5«  —  6 ;    a;3  4-  6.r2  +  112;  +  6 ; 
.'"  4-  4.r2  4-  x  —  6. 


CHAPTER    VII. 

FRACTIONS. 

174.  Fractions  in  Algebra  are  expressions  for  division, 
in  which  the  dividend  is  written  over  the  divisor  with  a  line 
between  them,  called  the  dividing  line. 

175.  The  Numerator  is  the  quantity  above  the  line, 
or  the  dividend. 

176.  The  Denominator  is  the  quantity  Mow  the  line, 
or  the  divisor. 

177.  The  Terms  of  a  fraction  are  its  numerator  and 
denominator. 

178.  A  fraction  is  in  its  lowest  terms  when  the  numer- 
ator and  denominator  have  no  common  factor. 

179.  The  Value  of  a  fraction  is  the  quotient  of  the 
numerator  divided  by  the  denominator. 

180.  An   Integral  or    Entire    Quantity    is  one 

expressed  without  fractions. 

181.  A  Mixed  Quantity  is  one  which  has  an  integral 
and  a  fractional  part. 

182.  Fractions  whose  denominators  are  alike  are  said  to 
have  a  Common  Denominator. 

183.  A  Compter  Fraction  is  one  having  a  fraction 
in  the  numerator  or  denominator,  or  both. 

Note. — The  terms  proper,  improper,  simple,  and  compound,  when 
applied  to  algebraic  fractions,  have  the  same  meaning  as  in  Arithmetic. 


64  FKACTIONS. 


SIGNS      OF      FRACTIONS. 

184.  Every  Fraction  has  the  sign  +  or  —  expressed 
or  understood  before  the  dividing  line;  that  is,  when  its 
direction  is  considered.     (Art.  86,  i  .) 

185.  The  Dividing  Line  has  the  force  of  a  vinculum 

or  parenthesis,  and  the  sign  before  it  belongs  to  the  value  of 
the  fraction. 

186.  Every  Numerator  and  Denominator  is 

preceded  b}r  the  sign  +  or  — ,  expressed  or  understood.  In 
this  case,  the  sign  affects  only  the  single  term  to  which  it  is 
prefixed. 

187.  If  the  sign  before  the  dividing  line  is  changed,  the 
sign  of  the  fraction  is  changed. 

Thus,  a  +  —  —  a  +  6,     but     a =  a  —  b. 

x  x 

188.  If  all  the  signs  of  the  numerator  are  changed,  the 
sign  of  the  fraction  is  changed. 

_.  +ax  —ax 

Thus.     =  +a,    and ■  =  —a. 

x  x 

139.  If  all  the  signs  of  the  denominator  are  changed,  the 
sign  is  also  changed. 

m  aX  x.    .        aX  tj 

Thus,       —  =  +a;    but    =  —a.    Hence, 

+  x  —x 

190.  If  any  two  of  these  changes  are  made  at  the  same 
time,  the  sign  of  the  fraction  will  not  be  changed. 

Thus,    —  =  +a.     Changing  the  signs  of  both  numerator  and  de- 

— ax 

nominator,      — -  =  +a. 

None.  —  By  the  application  of  the  above  principles,  tbe  sign  of  either 
term,  or  thai  before  the  dividing  line,  may  be  made  plus,  if  desirable. 


REDUCTION     OF     FRACTIONS.  65 


PRINCIPLES. 

191.   The  principles   for  the   treatment  of   fractions  in 
Algebra  are  the  same  as  those  in  Arithmetic. 

i°.  Multiplying  the  numerator,  or   1    Multiplies  the  frac- 
Dividing  the  denominator,  \  tion. 

Thus,     rM  =  2-        And     \        =2- 
6  63  6-^-2       3 

20.  Dividing  the  numerator,  or        ) 

,,  7, .  7   .      ,7     ,  •     ,  \  Divides  the  fraction. 

Multiplying  tlie  denominator,      \  •' 

m  2_H2        *  a    j      2  2         1 

Thus,    -        =  -  •       And     -        =  —  =  -  • 
6  6  6  x  2       12       6 

30.  Multiplying  or  dividing   both  )    Does  not  change  its 
terms  by  the  same  quantity     )  value. 

m  2X2  4  2         1  2-4-2         I 

Thus,    ; =  —  =-  =  -•       And     - =  -• 

6x2       12       6       3  6-=-2       3 


REDUCTION      OF      FRACTIONS. 

192.   Reduction  of  Fractions  is   changing  their 
terms  without  altering  the  value  of  the  fractions. 


CASE      I. 

193.    To  Reduce  a  Fraction  to  its  Lowest  Terms* 

r  (fib^dois 
1.  Eeduce  — — 5 —  to  its  lowest  terms. 
i$abcx 

Analysis. — By  inspection,  we  perceive  the  factors  5,  a,  b,  and  x  are 
common  to  both  terms.     Cancelling  these  common  factors,  the  fraction 

becomes    —  •    Now  since  both  terms  have  been  divided  bv  the  same 

3^ 
quantity,   the   value   of  the   fraction   is   not   changed.     (Art.   191,  30.) 

And  since  these  terms  have  no  common  factor,  it  follows  that  —      arc 

the  lowest  terms  required.     (Art.  178.) 


66  REDUCTION     OF     FRACTIONS. 

NOTE. — It  will  be  observed  that  the  factors  5,  a,  b,  and  x  are  prime  ; 
therefore,  the  product  $abx  is  the  </.  ft  ll.  of  the  numerator  and  denomi- 
nator.    (Art.  164.)     Heuce,  the 

Eule. — Cancel  all  the  factors  common  to  the  numerator  and 
denominator. 

Or,  Divide  both  terms  of  the  fraction  by  their  greatest 
common  divisor.     (Art.  178.) 

Eeduce  the  following  fractions  to  their  lowest  terms  : 
2jazb2c5d  x2  -f  y2 


62a5cid3e  x6  —  y6 


x  —  42 


3'      gaWcT  7'     ^+Q.r+U 

zs  +  y*  Q      4^2  —  4 

4-        75 Th'  «• 


x2  —  y2  6x*  —  6 

x6  4-  1  3a2  —  6«a;  4-  32^ 


J      or1  —  1  y  $x2  —  3a2 

CASE      II. 

194.    To  Reduce  a  Fraction  to  an  Entire  or  Mixed  Quantity. 

Since  a  fraction  is  only  an  expression  for  division,  we 
have  the 

Rule. — Divide  the  numerator  by  the  denominator. 

Reduce  the  following  to  entire  or  mixed  quantities  : 

4x2yjz?  a18  4-  b18 

2x2yh2'  5*      «6  4-  br 
b3  —  aW  aPx*  4-  2«2z3  4-  ax2 

1  —  a2  a2x2  4-  ax 

a8  —  bz  x3  4-  2x2  —  \2X  —  13 


3- 


a  —  b  x2  4-  x  —  1 2 

«10  +  bw  a2b2  -b2  +  ab-b-a  +  i 


a2  4-  V  '  a2 -1 


KEDUCTION     OF     FEACTION8.  67 


CASE      III. 
195.   To  Reduce  a  Mixed  Quantity  to  an  Equivalent  Fraction. 

i.  Keduce  a  -f-   -  to  the  form  of  a  fraction. 

3 
Analysis. — Since  in  i  unit  there  operation. 

are  three  thirds,  in  a  units  there  must  b         XCl        b 

a  +  -  =  - — f-  - 

be    a    times    3    thirds,  or    — ;    and  3  3         3 

3 

^  +  b  = 
3        3  3'  33  3 


-  = -,    the     fraction    re-  H  -  =  >  A^S. 

3  3 

quired.     Hence,  the 

Rule. — Multiply  the  integral  part  by  the  denominator  ;  to 
the  product  add  the  numerator,  and  place  the  sum  over  the 
denominator . 

Note. — An  entire  quantity  may  be  reduced  to  the  form  of  a  fraction 
by  making  1  its  denominator.     Thus,  a  =  -  • 

Reduce  the  following  to  the  fractional  form : 

2.  ax2 5.     a  —  x  —  - —  — ■-• 

a  a  —  x 

x  —  a2b2  ,  7       a2  +  b2 

3.  ab  + r 6.     a  +  b —r- 

ab  a  +  b 

(x—1)2  „        „  a4— 1 

4.  ic  +  1  +  - —  — ■ — •  7.     a3  +  ^  +  a;  +  1 —  • 


CASE      IV. 
196.   To  Reduce  a  Fraction  to  any  required  Denominator. 

By  Art.  191,  30,  we  have  the 

Rule. — Multiply  both  terms  by  the  factor  which  will  give 
the  fraction  the  required  denominator. 

Notes. — 1.  An  entire  quantity  may  be  reduced  to  a  fraction  having' 
a  given  denominator,  by  the  same  rule,  by  first  writing  under  it  the 
denominator  i. 

2.  When  the  required  denominator  is  not  a  multiple  of  the  given 
denominator,  the  result  will  be  a  complex  fraction. 


68  REDUCTION     OF     FRACTIONS. 


EXAMPLES. 

tt2  -\-  2 

i.  Reduce  -  — ■    to   a  fraction    whose    denominator    is 
c 

a2c  —  c. 

2.  Reduce  —n to  a  fraction   whose    denominator   is 

a2  +  i 

a6  +  i. 

2.  Reduce  -       -  to  a  fraction  whose  denominator  is  «4 — i. 
a  —  i 

X  4-  C 

4.  Reduce  -    — -    to   a  fraction   whose    denominator    is 
x  —  7 

X2  —  2X  —  35. 

<.  Reduce  -= to   a  fraction   whose    denominator    is 

D  a2  —  1 

a3  —  1. 

6.  Reduce  x to  a  fraction  whose  denominator  is 

x  +  1 

x2  —  1. 

CASE     V. 

197.  To  Reduce  Fractions  to  a  Common  Denominator. 

By  Case  IV  we  can  reduce  fractions  to  any  required 
denominator;  but  to  avoid  complex  fractions  that  denomi- 
nator must  be  a  multiple  of  the  given  denominators. 

To  express  the  fractions  in  the  lowest  terms,  it  must  be  the 
least  common  multiple.     We  have  then  the  following 

Rule. — I.  Find  a  common  multiple  of  the  denominators  for 
the  common  denominator,  the  least  common  multiple  being 
preferred. 

II.  Multiply  loth,  terms  of  each  fraction  by  that  factor 
which  will  give  it  the  required  denominator. 

198.  When  the  least  common  multiple  is  taken  as  a 
common  denominator,  it  is  called  the  Least  Common 
Denominator, 


ADDITION     AND     SUBTRACTION.  (i!) 

Eeduce    the    following    fractions    to   the    least  common 
denominator : 


2. 

3- 

4- 

5- 
6. 


x  +  y' 

i 

JC4—  i' 

a  +  i 

2a2  +  2 

4 

x2  -  1 ' 

a; 

.T*  -   I  ' 

.r  +  y 

X2 

a4 

27* 

x*  —  y2' 

"^ 

2 

3 

x*+  1' 

z3 

+  1 

a  —  1 

a2  —  1 

4«3  —  4 ' 

8a6  — 8 

x  —  2 

2' 

x  +  2 

a2  +  x  — 

a;2  —  £  —  2 

X2  -f   I 

x2—  1 

xA  +  4Z2  +  3 '      a4  +  2£2  —  3 
x  —  y         x2  +  y2 
xA  —  y*'      x2  +  y3 '      a;2  —  ?/3 


ADDITION    AND     SUBTRACTION     OF    FRACTIONS. 

199.  According  to  Art.  113,  fractions  may  be  added  or 
subtracted  by  the  following 

Kule. — Reduce  the  fractions  to  a  common  denominator,  and 
add  or  subtract  their  numerators,  writing  the  result  over  the 
common  denominator. 


2ax  3a 

1.  Add  — =-  and  - — 

30  2X 

Solution. — Reduced  to  a  c.  d.,  the  fractions  become  —  --  and  %r- , 

tax  6bx 

which,  by  adding  tbe  numerators,  give   — ~~2r~ >    Ans. 

2.  From  -—  take  - —  ■ 

x  y 

Solution. — Reducing    to   a  c.  d.,   we    have   - — -  and  -  — ;  and 

xy  .11/ 

"jaby      $cdx  _  yaby  —  scdx       . 
xy         xy  xy 


70  MULTIPLICATION     OF     FRACTIONS. 


Perform  the  operations  indicated  in  the  following: 
3- 


2<ix      a2  4  a2       x3  —  2a3 

+ 


52   ^        J2  ub*  +  b*z 

20b         a  —  b      a  4-  b 

4"    _     1    ~z     1 


a2  —  62      a  +  b      a  —  b 
4xy         x  +  y 


x  +  i 

1 

#  +  3  #2  —  # 

a2  4-  .c2      rr3  +  a*x        —  a?x 

x  4-  a2       a:2  —  a4       a;  —  ft2 

a;  a-  1 

a*2  —  9  a?  —  a  —  6       a'2  4-  Sx  +  ^ 
—  1  —  1  —  1  —1 


x  —  1       x  4-  1       a*2  —  1       a'3  —  1 

200.  The  sum  of  the  numerators  of  several  equal  fractions, 
placed  over  the  sum  of  the  denominators,  forms  a  fraction 
equal  to  one  of  the  given  fractions;  for,  if  they  be  reduced 
to  a  c.  d.,  they  will  be  identical,  and  the  operation  is 
equivalent  to  multiplying  both  terms  of  one  fraction  by  the 
number  of  fractions. 


MULTIPLICATION     OF     FRACTIONS. 
CASE    I. 
201.  To  Find  the  Product  of  an  Integral  Quantity  and  a  Fraction. 

According  to  Art.  191,  i°,  we  have  the  following 

Eule. —  Cancel  all  factors  common  to  the  integral  quantity 
and  the  denominator,  and  multiply  the  numerator  by  the 
remaining  factors  of  the  integral  quantity. 

Notes. — 1.  A  fraction  is  multiplied  by  any  factor  by  cancelling 
that  factor  from  its  denominati  r. 

2.  Cancelling  the  tchole  denominator  multiplies  the  fraction  by  the 
denominator. 


MULTIPLICATION     OF     FRACTIONS  71 

Find  the  product  of  the  following: 

i.     -3 „  x  (a  —  x). 

a2  —  x2       K 

2. x  (x2  +  1)  (x  —  1). 


3.  (iC4  +  X3  +  .T2  -f  X  +    I )    X    -y 

4.  (*+!»-  l)    X^-- 

X  +  A  ,  s 

5-        o    ,        — X    («  +  7). 

a-2  +  4.T  —  2 1        v  7 

x 


6.     (z2  -f  5.?  +  6)  x 


7.     (.T2  +  a5)  x 


:Y3  +  gx2  -f-  26a;  +  24 


a;4  —  «10 
10 


a;13  +  a3 


X  (z4  +  a). 


CASE      II. 
202.    To  Multiply  a  Fraction  by  a  Fraction. 
a  ,      ra 


1.  Multiply  r  by  — 
1  J   b     J    n 


Solution. — By  Arts.  53  and  54,  the  product  may  be  written 

am  am 

-  x  — ,  or  ub-1  x  mn ■    ,  or  ab~  1mn-1.  or  - — • 

0      n  on 

If  in  this  result  there  are  common  factors  in  the  numerator  and  denom- 
inator, they  might  hare  been  cancelled  before  multiplying.     Hence,  the 

Eule. — I.  Cancel  all  factors  of  the  numerators  and  denom- 
inators common  to  both. 

II.  Multiply  the  remaining  factors  of  the  numerators  for 
the  numerator  of  the  product,  and  the  remaining  factors  of 
the  denominators  for  the  denominator  of  the  product. 

Note. — This  rule  applies  to  any  number  of  factors. 


72  DIVISION     OF     Fit  A  C  TIOXS. 

Find  the  product  of  the  following: 
a  a  +  x 


4- 

5- 


a2  —  x1  a 

x3  a  —  b2 


"        «2  _  £4  X2 

3. I x-3_-.  6: 

0      a2  +  a;  +  1       x  —  1  a? 

«2  4-  «  —  42       a:2  —  a;  —  20 

7.  — ' —  x • 

x2  +  x  —  30       «'-  —  a  —  30 

a  4-  1        r2  —  1       «  —  1 

8.  -r— —  x  -3- —  x 

.'-3  +1       aA  +  1       a;  —  1 

'    av          x*  —  1            a;2          re4  4-  a;2  4-  1 
x  — s—  x x  -  


1 

X 

a2 
a2 

4-  J3 

a4 

— 

bG 

-P 

a;2 

+ 

a? 

X 

X9 

.'■'' 

4-  ft6 

.r:i 

+ 

a2 

4- a9 

a;3 

— 

a5 

V 

X* 

»-a6 

a;2  4-1  x2  x  —  1 

DIVISION      OF     FRACTIONS. 

CASE     I. 

203.    To  Divide  a  Fraction  by  an  Integral  Quantity. 

By  Art.  188,  20,  we  have  the 

Rule. —  Cancel  from  the  numerator  of  the  fraction  and  the 
divisor  all  common  factors,  and  multiply  the  denominator  by 

tlii  remaining  factors  of  t  lie  divisor. 

Divide  the  following  : 

a2bc~  0  1    1  8xh/»z2  5011 

1.     — ; j-  orc*x*,  2.     — r^j =-  4a'*csz-ysc 

$x?yz2  ga*c»d 

a2x?  —  air.       r  „  .  .   .  .  _ 

4-         — •-  [a2  (x  4-  a)  (x  -  a)]. 

8  + 6a  +  a2  ,  . 

5. J ; 2  -v-  (0  —  5«  —  2ft2  4-  a3). 

34-  4ft  4-  ft-        v  ' 

6-  irpf  -*-  [«  -  (fl  +  *)  y  +  art- 


DIVISION      OF     FRACTIONS.  73 

CASE      II. 
204.    To  Divide  a  Fraction  by  a  Fraction. 


i.  Divide  j  by 


m 
n 


Solution.— By  Art.  54  tliis  may  be  written 

am  ,    ,  ab-'  am,-1  an 

=-  -s —  ,     or    ab~[  -J-  vm-,     or        — ,  ,      or      ^ ,     or     - —  • 

0       n  mn—1  on,-1  bm 

From  the  last  two  expressions  we  derive  the 

Rule. — Divide  the  terms  of  the  dividend  by  the  correspond- 
ing terms  of  the  divisor. 

Or,  Multiply  the  dividend  by  the  divisor  inverted. 

These  operations  may  be  combined  in  the  following 

Eule. —  Cancel  all  factors  common  to  both  numerators,  also 
those  common  to  both  denominators,  and  midtiply  the  dividend 
by  the  divisor  inverted. 

205.  To  Divide  an  Integral  Quantity  by  a  Fraction. 

Make  the  integer  a  fraction  by  writing  i  for  a  denominator, 
and  proceed  as  above. 

1.  Divide  (5a  —  b)  by  — 

2.  Divide  (1  —  x%)  by 

J      3« 

206.  Complex  Fractions  are  reduced  to  simple  ones 
by  performing  the  division  indicated. 

Divide  the  following: 

2a2z*y    i     ax*y%  c15  —  x13   %   c5  —  x6 


1. 


6bsc?d2   '  &&d*e  a*  +  x6    '  di  +  z2 

c6  —  d6      p    a<?  —  ad3 
a*b*  —  ~(W  "*"  aV  —  WcH 

yu  +  1  ^        y4+  1 

a6  —  1       a3  +  a2  —  a  —  1 " 
4 


74 


FRACTIONS. 

5- 

XU 

a~ 

ah* 

4-  a2y  2 

a~ 

+  y~l 

A 

r' 

—  $c  — 

14    . 

c2  +  HC  — 
c2  +  c  — 

-  26 

c* 

—   IOC  — 

■39 

6 

1± 
16 


a4  —  2d 


aP  —  y21 

a16  —  ?/2° 

a3  —  y1 

a1  +  2a  —  8  (a8  +  yw)  (a*  +  y*) 


207.  The  various  rules  for  multiplication  and  division  of 
integers  and  fractions,  whatever  mav  be  the  number  of  multi- 
pliers  and  divisors,  may  be  summed  up  in  the  following 


GENERAL   RULE. 

I.  Reduce  integers  and  mixed  quantities  to  fractions. 
II.  Invert  all  the  divisors. 

III.  Cancel  all  the  factors  common  to  the  numerators  and 
denominators. 

IV.  Combine  the  remaining  factors  for  the  result. 

208.  From  the  nature  of  fractions  we  readily  see  that  the 
value  of  a  fraction  may  be  any  quantity  whatever  ;  for  as  the 
denominator  dmrasrs,  with  a  given  numerator,  the  value  of 
the  fraction  increases,  and  when  the  denominator  becomes 
infinitely  small,  or  infinitesimal  (a  quantity  represented  by  o), 
the  value  of  the  fraction  becomes  infinite. 

209.  On  the  other  hand,  when  with  a  given  denominator 
the  numerator  decreases  indefinitely,  the  value  of  the  fraction 
decreases  and  becomes  infinitesimal  or  o.  Hence  we  have  the 
equations, 

-  =  co.  (i)  _  =  o.  (3 

O  cc 

y  =  °-  (2)  I  =  a-  (4) 

a  o 


FRACTIONS.  75 

210.  These  are  important  relations  for  the  student  to 
remember.     Translated  into  common  language  they  become: 

ist.  A  finite  quantity  divided  by  zero  equals  infinity. 

2d.  Zero  divided  by  a  finite  quantity  equals  zero. 

3d.  A  finite  quantity  divided  by  infinity  equals  zero. 

4th.  Zero  divided  by  zero  equals  any  quantity. 

Equation  (4)  is  called  indeterminate  because,  as  it  stands,  its  value 
cannot  be  determined.  It  is  only  when  the  functions  are  known  from 
which  the  two  zeros  have  resulted  that  a  definite  value  can  be  given  to 
such  a  fraction,  and  then  only  when  these  functions  are  dependent.     For 

example,  if becomes  -  by  reason  of  x  becoming  equal  to  a  and 

y  to  b,  the  resulting  -  is  indeterminate,  and  such  an  answer  to  a  problem 

must  be  interpreted  as  meaning  that  the  problem  has  no  definite  answer, 
but  its  conditions  are  satisfied  by  any  value  whatever  for  the  required 
quantity. 

t,      .  „   a  (a*  —  x")   ,  o  ,  „       , 

13ut  11   — become  -  by   reason   or   x  becoming  equal  to  a, 

,V  (ft        tC)  o 

we  may  find  a  definite  value  for  the  expression  by  first  reducing  the 
fraction  to  its  lowest  terms,  thus, 

a  (a2  —  x9)  _  a  (a  +  ir) 
x  (a  —  x)  x        ' 

which  when  x  =  a,  becomes  2a. 

Perform  the  operations  indicated  in  the  following,  and 
simplify  the  expressions : 


I. 

a  +  b 
a-b^ 

a  —  b         a-  —  b2 
a+b      a  (a  —  b) 

2. 

a  —  1 

+ 

a 

a 

a  —  1 

a3  —  Xs 

a2  —  x2 

3- 

a2  —  x2 

a  —  x 

"1'  + 

a  +  1 

a  -f-  1       a2  —  1 

4- 

a  —  1       a2  +  1 

5- 

a  —  1 

X 

a  +  1 

a  +  1    t  «2  —  1 

a  —  1    "    a  +  x 

6. 

xz  —  «3 

x  —  a 

x2  —  di 

x  +  a 

76 


7.     a  +  x 


FRACTIONS. 

a2  +  X2 
a  +  x 


a  —  x      b  —  x      x  — 
ax  ab  bx 


i  i 

+ 


x  —  I 


X  —  I  X  +    I  X2  —  I 

I    ai  —  .r4       a2  +  x2 


ii. 


x2  —  if        x  -j-  y 

i  i 

a  +  x       a  —  x 

a?  +  3«2#  +  $ax2  +  x3 
a2  +  2ax  +  x2 

(-3(»-J) 


2a;. 


I3'     \a  -f  &  +  a  —  b)  '  \a  —  I      a  +  d/ 

la2  +  ff        \  «2  —  52 
\      J  ft/  «3  +  S8 


IS- 


1 6. 


a:  — 


z  + 


[7.     (a3  —  3a2  +  3«  +  1)  -=-  (rt2  —2C1  +  ^—jj' 


18. 


2/       «    .   (x  +  #)2  —  2a3/ 


*  —  # 


CHAPTER    VIII. 

EQUATIONS     OF     THE     FIRST     DEGREE. 

211.  Algebra  has  already  been  defined  as  the  Science 
of  the  Equation.  The  preceding  chapters  have  been 
devoted  to  the  fundamental  operations  upon  quantity  in 
preparation  for  the  reduction  and  discussion  of  equations. 

212.  An  Equation  is  an  expression  of  equality  between 
two  quantities;  as,  a  =  b  +  c.     (Art.  26.) 

213.  These  two  quantities  are  called  Members  of  the 
Equation,  the  one  on  the  left  of  the  sign  =  being  the 
First  Member,  and  the  one  on  the  right  the  Second 
Member. 

214.  Equations  are  Literal  when  the  unknown 
quantities  have  literal  coefficients,  and  ^Numerical  when 
their  coefficients  are  numerical. 

Thus,  ax2  +  bx  =  c  is  a  'literal  equation,  and  3a:2  —  2X  —  10  is  a 
numerical  equation. 

215  Equations  are  of  different  degrees,  depending  on  the 
exponents  of  the  unknown  quantity. 

To  determine  the  degree,  these  exponents  must  be  expressed 
in  the  same  unit ;  that  is,  must  have  a  common  denominator. 

The  degree  will  then  be  found  by  subtracting  the  least 
from  the  greatest  exponent  of  the  unknown  quantity;  or,  if 
there  are  no  negative  exponents,  the  degree  will  be  equal  to 
the  greatest  exponent. 

Thus,  x-  —  ar-1  =  5  is  of  the  third  degree  ;  x  +  -  =  7  is  of  the  second 
degree ;  and  x2  +  x^  —  2  is  of  the  *  degree. 


78  EQUATIONS     OF 

216.  An  Identical  Equation  is  one  whose  members 
are  identical,  or  may  be  made  so  without  changing  then- 
value  ;  as, 

a  +  x  =  a  -f-  x; 

(a  +  x)'2  =  a2  -f-  2  ax  +  £2. 

217.  Such  an  equation  is  called  absolute,  since  its  truth 
does  not  depend  on  the  value  given  to  any  of  the  quantities. 

218.  A  Conditional  Equation  is  one  which  is  true 
only  on  certain  conditions  ;  as, 

x  —  2  =  5, 

which  is  true  only  when  x  =  7. 

219.  Such  equations  are  used  in  the  solution  of  problems. 
The  conditions  of  the  problem  are  expressed  by  the  equation, 
in  which  some  letter,  as  x,  is  put  for  the  unknown  quantity. 
Whatever  value  of  x  will  make  tbe  equation  true,  will  there- 
fore satisfy  the  conditions  of  the  problem. 

220.  A  Root  of  a  Conditional  Equation  is  that 
value  of  the  unknown  quantity  which  will  satisfy  the  equa- 
tion; as, 

X  -f  5   =   2X, 

in  which  5  is  a  root,  since  in  substituting  it  for  x  we  have 
5  +  5  =  2x5. 

REDUCTION     OF     EQUATIONS. 

221.  The  Reduction  of  an  Equation  consists  in 
such  transformations  as  will  make  the  unknown  quantity  an 
explicit  function  of  the  known  quantities.     (Arts.  3$,  34.) 

Thus,  in  the  equation, 

a  +  x      a  —  x 


b  c     ' 

a;  is  a  function  of  a,  b,  and  c  ;  that  is,  the  value  of  x  depends  on  the  values 
given  to  a,  b,  and  c.  This  is  implied  in  the  equation  ;  for  to  affirm  that 
any  combination  of  a!  with  a,  b,  and  c  is  equal  to  a  different  combination 
of  the  same  quantities  is  to  make  each  one  of  the  quantities  depend  oil 


THE     EIRST     DEGREE.  79 

the  others  for  its  value.  If  all  but  one  have  arbitrary  values  assigned 
them,  the  remaining  one  cannot  have  such  a  value  assigned  to  it  and 
make  the  equation  true ;  but  it  must  have  a  definite  value,  which  will  be 
expressed  by  some  combination  (that  is,  some  function)  of  those  quan- 
tities to  which  the  arbitrary  values  have  been  assigned. 

222.  Those  quantities  to  which  arbitrary  values  may  be 
assigned  are  called  Known  Quantities,  and  those  which 
are  to  be  found  by  their  relations  to  the  known  are  called 
Unknown.    (Arts.  27,  28.) 

223.  In  the  equation  above,  if  we  let  a,  b,  and  c  represent 
known  quantities,  the  equation  implies  that  £  is  a  function 
of  a,  b,  and  c,  but  does  not  explicitly  state  what  function  it  is. 
When  the  equation  is  reduced,  it  becomes 

ab  —  ac 

x  =  —= , 

b  +  c 

which  states  explicitly  what  function  of  a,  b,  and  c,  equals  x. 

224.  This  reduction  is  effected  by  the  axiom, 

Equal  quantities  equally  affected  remain  equal.     (Art.  38.) 

225.  The  Method  of  Reduction  depends  on  the 
degree  of  the  equation.     (Art.  215.) 

226.  Equations  of  the  First  Dcr/rcc  are  also 
called  Simple  Equations,  and  can  contain  only  two 
powers  of  the  unknown  quantity.  These  powers  (unless 
there  are  negative  exponents)  will  be  the  first  and  the  zero 
power. 

Xote. — Let  the  student  observe  that  the  term  which  does  not  contain 
the  unknown  quantity,  commonly  called  the  absolute  term,  is  said  to  con- 
tain the  zero  power  of  that  quantity. 

227.  The  lied  action  of  equations  of  the  first  degree  is 
effected  by  the  following  operations: 

1  st.  Clearing  of  fractions ;  that  is,  removing  denominators. 
2d.  Kemoving  known   terms  from  the  first  member  and 
unknown  terms  from  the  second  member. 
3d.  Uniting  similar  terms. 


80  E  Q  U  A  T 1 0  X  S     0  F 

4th.  Making  the  coefficient  of  the  unknown  quantity 
unity. 

Note. — These  operations  may  be  performed  in  any  order  which  is 
most  convenient. 

228.  To  Clear  an  Equation  of  Fractions. 

Rule. — Multiply  the  equation  by  the  least  common  multiple 
of  its  denominators. 

229.  To  Remove  a  Term  from  either  Member. 
El'LE. — Subtract  the  term  from  both  members. 

230.  To  Make  the  Coefficient  of  the  Unknown  Quantity  Unity. 
Rule. — Divide  both  members  by  this  coefficient. 

EXAMPLES. 

231.  Find  the  value  of  x  in  the  following  equations: 

■xx  +  ia      x  —  5« 
2  3 

Operation. — Multiplying  by  6, 

gx  +  6a  —  2X  +  10a  =  30a. 

Subtracting  6a  and  10a  from  both  members,  and  uniting  similar  terms, 

"jx  =  14a. 
Dividing  by  7,  X  —  2a,     Ans. 

Note. — It  will  be  observed  that  the  removal  of  a  term  from  one 
member  of  an  equation  causes  it  to  appear  in  the  oilier  member  with  an 
opposite  sign. 

This  is  called  Transposition  ;  because  it  is  equivalent  to  trans- 
posing a  term  from  one  member  to  the  other  and  changing  its  .sign. 

232.  Many  equations  not  of  the  first  degree  may  be  made 
such  by  some  operation  affecting  equally  both  members.    Thus. 

2.  Reduce  x2  —  4  =  2  (x  +  2). 

Dividing  by  x  +  2,  X  —  2  =  2. 

Transposing,  x  =  4,     Ans. 


THE     FIRST     DEGREE.  81 

This  equation  is  of  the  second  degree,  but  by  dividing  both  members 
by  X  +  2  it  is  reduced  to  the  first  degree,  and  4  found  to  be  a  root. 

But  since  both  members  are  divisible  by  x  +  2,  any  value  of  x  that 
will  make  this  factor  zero  will  satisfy  the  equation,  for  it  will  reduce 
both  members  to  zero.  * 

Thus,  x  +  2  =  o        gives        x  =  —  2. 

.•.    —  2  is  another  root  of  the  equation. 

We  shall  see  hereafter  that  every  equation  has  as  many  roots  as 
there  are  units  in  its  degree,  and  that  when  we  divide  by  a  factor  con- 
taining x,  we  must  make  another  equation  by  putting  that  factor  equal 
to  zero,  and  find  its  roots,  if  we  would  find  all  the  roots  of  the  original 
equation. 

3.  Reduce  (x  +  i)»  =  —  2. 
Squaring  both  members,  x  +  1  =  4. 
Transposing,  etc.,  x  =  3. 

233.    To  Prove  the  Correctness  of  the  Work  of  Reduction. 

Substitute  the  root  found  for  the  unknoion  quantity  in  the 
equation.     If  correct,  it  ivill  reduce  to  an  identical  equation. 

Thus,  in  the  last  example,  substituting  3  for  x,  we  have, 

(3  +  1)*  =  —  2. 
But  (3  +  1)*  =  4*  =  +  2  or  —  2. 

While,  therefore,  the  equation  is  satisfied  by  the  root  3  if  the  nega- 
tive root  of  4  be  taken,  it  is  not  satisfied  if  the  positive  root  be  taken. 

But  the  equation  is  of  the  i  degree,  and  if  the  statement  in  Art.  232 
be  true,  the  root  found  should  be  a  half-root,  as  we  see  it  is ;  that  is, 
it  satisfies  the  equation  only  in  one-half  the  ways  in  which  it  can  be 
substituted. 

_  ,        x  +  a      x  —  a 

4.  Reduce  — 7 1 =■ —  =  10. 

In  this  equation  we  may  unite  the  terms  with  advantage  before 
clearing  of  fractions  ;  thus, 

2X 

T  =  io. 

2 
Dividing  by      ,  we  have       x  =  56,    Ans. 

5.  Reduce  ax  -+-  ix  =  c. 

Dividing  by  a  +  b,  x  =  -.  ,     Ans. 

a  +  0 


82  EQUATIONS     ()!■ 

Reduce  the  following,  and  prove  the  work  by  substituting 
the  root  found : 

x       x       x       x  +  2 

234        2 

x  —  1       x  —  3  x  —  3 

242 
a  —  x       b  —  x       c  —  x 


-     13 
14 

15 
_      16 

17 

0 

-     18 

19 

20 
21 


ax2  —  bx  =  bx2  —  ex. 

(a  4-  x)  (b  +  x)  =  (m  +  x)  (n  4-  x). 

x  —  a       x  —  b       x  —  c  x  —  (a  4-  Z>  +  c) 

b  c  a  abc 


x  —  2       x  —  4        x  —  6       x  —  8 
?»  (./•  4-  a)       n  (x  4-  b) 
a-  4-  b  x  4-  « 

X  4-  S 2  —12 Z. 

2  3 

a;  —  4  ca;  4-  14        1 

3X 4  - 


5  —  6x  + 


3  12 

7a:  4-  14        17  —  3a;      4a-  4-  2 


3 


?o  —  a;       6a;  —  8       4a;  —  4                xx  —  x 
+  - =  x  —  2 *  +  4, 

2  7  5  5 

zs  +  4  _  6a:  4-  7       7a  —  13, 

3  9  6a;  4-  3  ' 


(x  —  b\  (x  +  b\       x 


3     /  v 

a;  h  -        x 


a  4-  1  a  —  1 

c  a;  2  4-  a; 

a2  —  #2       a  —  b      a  +  b 

x  —  2    ,    .301  „       .r  —  2 

h  — —  =  .ooia-  4-  .6  —    

5  -5  -05 


THE     FIRST     DEGREE.  83 


SOLUTION     OF     PROBLEMS. 

234.  The  Solution  of  a  Problem,  which  requires 
the  finding  of  an  unknown  quantity  from  its  relations  to 
known  quantities,  consists  of  three  distinct  steps.     (Art.  30.) 

1st.  The  conditions  of  the  problem  must  be  expressed  hy  a 
conditional  equation. 

2d.  That  equation  must  be  reduced  from  an  implicit  to  an 
explicit  function,  called  a  formula. 

3d.  The  numerical  values  of  the  known  quantities  must  be 
substituted  in  that  formula. 

235.  The  first  of  these  steps  does  not  belong  to  Algebra  ; 
but  as  the  practical  value  of  Algebra  cannot  be  illustrated 
except  by  the  solution  of  problems,  it  is  important  to  become 
familiar  with  the  translation  of  the  conditions  of  a  problem 
into  an  equation. 

The  problems,  however,  which  may  properly  engage  the 
attention  of  the  student  are  those  relating  to  subjects  with 
which  he  is  supposed  to  be  familiar. 

236.  This  analysis  of  the  process  of  solving  a  problem 
gives  the  following 

Rule. — I.  Represent  the  quantities  involved  hy  proper 
letters,  in  accordance  with  the  usage  of  the  algebraic  language. 
(Arts.  41-69.) 

II.  With  these  letters  express  the  conditions  of  the  problem 
by  a  conditional  equation. 

III.  Reduce  this  equation.     (Arts.  224-227.) 

IV.  To  apply  the  solution  to  a  special  case,  substitute  the 
numerical  values  given  in  the  special  case,  in  the  formula 
obtained. 


84  EQUATION  SOF 

237.  The  following  problems  will  illustrate  the  rules: 

Problem  i.  Find  the  number  which  being  divided  by 
two  given  numbers  will  give  quotients  differing  by  a  given 
number. 

Solution. — ist.  Let  x  =  the  number  to  be  found. 
m  and  n  =  the  given  divisors. 

d  =  the  difference  of  quotients. 
Having  assumed  this  notation,  we  are  prepared  to  express  the  con- 
ditions of  the  problem  by  an  equation.     This  equation  will  be 

x       x  —  ii 
m      n  ' 

in  which  x  is  an  implicit  function  of  m,  n,  and  d. 

2d.  The  second  step  in  the  solution  is  the  reduction  of  this  equation, 
by  which  x  becomes  an  explicit  function  of  the  given  quantities.  This 
reduction  gives 

_      mnd 
n  —  m 

3d.  The  third  step  consists  in  applying  this  explicit  function  or 
formula  to  any  given  case  of  the  problem. 

2.  Find  a  quantity  whose  fifth  part  exceeds  its  sixth  part 
by  5- 

Here    m  —  5,     n  —  6,    and     d  =  5  ;     and 

x  =  — =  mo,  Ans. 

6-5 

238.  The  third  step  of  the  solution,  which  is  arithmetical. 
may  be  performed  before  the  second  by  substituting  in  the 
equation,  before  it  is  reduced,  the  numbers  belonging  to  the 
special  case. 

Thus,  -  -  -  =  5. 

5       6       5' 

Reducing,  x  =  150,    Ans. 

239.  The  advantage  of  reducing  the  equation  before  the 
arithmetical  substitutions  are  made  is  evident  from  the  fact 
that  a  formula  or  rule  is  obtained  by  which  the  arithmetical 
part  of  the  solution  may  be  performed  for  any  special  case  of 
the  problem. 


m  +  x 
n  +  x 

m  — 
x  = 

rn 

THE     FIRST     DEGREE.  85 


DISCUSSION     OF     FORMULAS. 

240.  A  problem  is  said  to  be  Generalized  when  its 

conditions  are  stated  in  general  terms  and  reduced  to  -a  formula. 
The  Discussion  of  a  Formula  consists  in  applying 
it  to  such  special  cases  of  the  problem  as  will  show  the  differ- 
ent forms  the  result  may  take. 

3.  Find  the  time  when  the  ages  of  two  persons,  A  and  B, 
will  have  a  given  ratio,  the  present  age  of  each  being  given. 

Let   x  =  the  time  required,   r  =  the  given  ratio,  m  =  A's  age,  and 
n  =  B's  age. 

Then,  by  the  conditions, 

and 

r  —  1 

To  discuss  this  formula  : 

1st.  Suppose  that  A  is  now  30  years  old  and  B  20.     How 
long  before  A  will  be  twice  as  old  as  B  ? 

In  this  case,     r  =  2,    m  =  30,     n  =  20. 

30  —  40       —  10 

.-.    x  = —  = =  —  10. 

2  —  1  +1 

This  means  that  the  event  occurred  10  years  ago. 

2d.  Again,  suppose  A  is  30  years  old  and  B  30.     How  long 
before  A  will  be  3  times  as  old  as  B  ? 

Here    m  =  30,     n  =  30,     r  =  3. 

r,   ,     -,    ,.  30  —  90       —  60 

substituting-,  x  =  —  = =  —  30. 

3-i  +2 

That  is,  30  years  ago,  when  the  age  of  each  was  zero,  A  might  be 
said  to  be  3  times  as  old  as  B.     0x3  =  0. 

3d.  Again,  let  A's  age  be  30  and  B's  30.     When  will  the 

ratio  between  their  ages  be  1  ? 

30  —  30      o 

Bv  the  formula,  x  = —  =  -  , 

1  —  1         o 

which,  by  (Art.  209),  is  indeterminate,  and  means  that  any  time  future  or 

past  will  satisfy  the  conditions  of  the  problem,  since  their  ages  are  now 

equal  (or  their  ratio  is  1),  and  have  been  and  will  continue  equal. 


86  EQUATIONS     OF 

4th.  Again,  let  A's  age   be  30,  and   B's  15.     How  long 
before  A  will  be  twice  as  old  as  B  ? 

Substituting,  x  =        — —  = =  o. 

2  —  1  +1 

That  is,  in  zero  time,  or  now,  their  ages  are  in  that  ratio. 

5 tli.  Once  more,  let  A's  age  be  30  and  B's  age  20.     How 
long  before  their  ages  will  be  equal. 


By  the  formula, 


_  30  —  20  _  +  10  _ 
1  —  1  o 


That  is,  only  at  the  end  of  an  infinite  time  ;  in  other  words,  they  will 
never  be  of  equal  age. 


PROBLEM     OF    THE     COURIERS. 

4.  Two  couriers,  A  and  B,  are  travelling  to  tbe'  cast,  the 
former  m  and  the  latter  n  miles  per  hour.  At  noon,  A  passes 
a  given  point  0,  and  B  is  a  miles  in  advance  of  A.  How  long 
after  noon  and  how  far  from  0  will  they  be  together? 

Let  t  —  the  required  time,  and  d  =  the  required  distance.     Then 
mt  —  nt  =  a, 

and  t  — • 

m  —  a 

ant 
~~  m  —  n 

To  discuss  this  result,  we  make  the  following  suppositions: 

1st.  Let  a,  m,  and  n  be  positive,  and  m  >  n. 

.".  t  and  d  are  positive,  and  the  time  of  meeting  is  after  noon,  and 

the  place  east  of  O.     t>  —  and  d  >  a. 
m 

2d.  Lei  a,  m,  and  n  be  positive,  and  in  <  v. 

.-.  t  and  d  are  negative,  and  the  time  is  before  noon,  and  the  place 
west  of  0. 

3d.  Let  a  =  o,  and  m  %  n  (read,  "  m  greater  or  less 
than  n"). 

r.  t  =  o  and  d  =  o.     The  time  of  meeting  is  noon  and  the  place  at  0. 


THE     FIRST     DEGREE.  87 

4th.  Let  m  =  o,  and  a  and  n  be  positive. 

:.  t  = and  d  =  o.     The  time  is  before  noon  and  the  place  as 

n 

before,  but  for  a  different  reason.     By  the  former  supposition,  A  passed 

I   O  at  noou  ;  by  the  latter,  he  remains  at  0  all  the  time. 

5th.  Let  n  =  o,  and  a  and  m  be  positive. 

.'.  t  =  —  and  d  =  a.     B  now  remains  at  a  miles  east  of  0,  but  the 
m 

time  is,  as  it  should  be,  —  hours  after  noon. 
m 


6th.  Let  m  =  n,  and  a  be  positive. 

.*.  t  =  oo  and  d  =  00 .  Their  rate  of  travelling  being  the  same,  it 
will  require  an  infinite  time  and  distance  for  A  to  overtake  B. 

7th.  Let  m  =  n,  and  a  =  o. 

.-.  t  =  - ,  and  d  =  -  •     Hence, 
o  o 

They  are  together  all  the  time  and  everywhere;  as  they  should 
be,  being  together  at  noon  and  travelling  at  the  same  rate. 

8th.  Let  a  and  m  be  positive,  and  n  negative. 

.•.  t  and  d  are  positive,  and  the  result  is  similar  to  1st,  but  t  <  — 

m 
and  d<a. 

9th.  Let  a  and  n  be  positive,  and  m  negative. 
.'.  t  is  negative  and  d  positive,  and  they  met  before  noon,  east  of  O. 

10th.  Let  a  be  negative  and  m  =  n. 

.'.  t  =  —  00  and  d  =  —  en  . 

That  is,  they  started  at  the  same  place  an  infinite  time  ago,  an 
infinite  distance  west  of  0  ;  in  other  words,  they  have  never  been  and 
never  will  be  together  in  any  finite  time. 

Note. — A  careful  examination  of  these  results  will  enable  the 
student  to  discuss  the  formulas  he  may  obtain  by  the  generalization  of 
problems. 

Let  the  student  generalize  such  of  the  following  problems 
as  are  not  made  general  by  the  statement. 


88  .   EQUATIONS     OF 


PROBLEMS. 

i.  A  man  has  3  times  as  many  half  dollars  as  he  has  dollars, 
5  times  as  many  quarters  as  he  has  halves,  3  times  as  many 
dimes  as  quarters,  and  5  times  as  many  half  dimes  as  dimes. 
The  whole  sum  is  44  dollars.     How  many  of  each  has  he  ? 

2.  In  a  family  of  six  persons  the  average  age  is  15  years. 
The  mother's  age  is  six  years  more  than  the  sum  of  the  chil- 
dren's ages,  and  the  father  is  six  years  older  than  the  mother. 
How  old  is  the  father? 

3.  On  a  certain  farm  there  is  twice  as  much  pasturage  as 
tillage,  and  33^  more  woodland  than  pasturage  and  tillage 
together.  If  40  acres  be  taken  from  the  pasturage  and  added 
to  the  tillage,  and  50  acres  from  the  woodland  and  added  to 
the  pasturage,  the  division  will  be  equal.  How  large  is  the 
farm  ? 

4.  The  sum  of  two  numbers  is  s,  and  their  difference  d. 
What  are  the  numbers  ? 

5.  Divide  a  into  two  parts,  such  that  their  difference  shall 
equal  ma. 

6.  Divide  a  into  two  parts,  such  that  the  difference  between 
one  part  and  b  shall  equal  n  times  the  difference  between  the 
other  part  and  c. 

7.  A  man  has  his  property  invested,  \  in  real  estate,  £  in 
government  bonds,  and  the  remainder  is  equally  divided 
between  stock  in  trade  and  money  loaned,  on  which  last 
investment  he  realizes,  at  6%  per  annum,  $900  a  year.  What 
is  the  value  of  his  whole  estate? 

8.  A  firm  doubled  its  capital  during  the  first  year  of 
business;  the  second  year  it  lost  8100  less  than  half  of  the  first 
year's  profits ;  the  third  year,  if  the  profits  had  been  $500  more, 
it  would  have  doubled  its  capital  again ;  as  it  was,  the  capital 
at  the  end  of  the  third  year  was  just  2\  times  the  original 
investment.     What  was  the  capital  at  first? 

9.  If,  in  going  a  journey,  a  horse  walks  half  way  at  the  rate 
of  3  miles  an  hour,  how  fast  must  he  go  the  remaining  half  to 
average  4  miles  an  hour?  How  fast  to  average  5  miles  an 
hour?     6  miles? 


THE     FIRST     DEGREE.  89 

10.  A  hoy,  a  years  ago,  was  one-half  as  old  as  his  mother; 
now  he  is  one-half  as  old  as  his  father.  How  much  older  is 
his  father  than  his  mother? 

ii.  A  merchant  sold  a  bill  of  goods,  one-half  at  a  gain  of 
33l'j'i  i  at  2°fo  an(l  the  rest  at  lo% ',  the  total  gain  was  $18.60. 
"What  was  the  amount  of  the  sale? 

12.  In  travelling  m  miles,  the  forward  wheel  of  a  carriage 
turns  n  times  more  than  the  hind  wheel.  If  c  represent  the 
circumference  of  the  forward  wheel,  what  is  the  circumference 
of  the  hind  wheel  ? 

13.  A's  age  is  -  of  the  sum  of  B's  and  C's,  and  the  sum  of 

all  their  ages  is  s.     What  is  A's  age  ? 

14.  A  can  reap  a  field  in  a  days,  B  in  b  days,  C  in  c  days. 
In  what  time  can  they  reap  it  together  ? 

15.  Half  the  contents  of  a  cask  containing  brandy  is  drawn 
off  and  20  gallons  of  water  put  in;  one-half  is  again  drawn  off 
and  50  gallons  of  water  put  in,  when  \  is  brandy  ?  How  much 
was  in  the  cask  at  first  ? 

16.  What  number  multiplied  by  a  and  the  product  added 
to  i  equals  c  ? 

17.  What  number  multiplied  by  m  gives  a  product  a  less 
than  n  times  the  number? 

18.  Two  men,  a  miles  apart,  travel  toward  each  other,  one 
m  miles,  and  the  other  n  miles  an  hour.  In  how  many  hours 
will  they  meet? 

19.  A  cask  holding  140  gallons  is  filled  with  brandy,  wine 
and  water.  There  are  10  gallons  more  wine  than  brandy,  and 
as  much  water  as  brandy  and  wine  together.  What  quantity 
is  there  of  each  ? 

20.  A's  earnings  for  the  past  year  are  Sioo  less  than  twice 
B's  ;  B's  are  $50  more  than  one-half  C's;  and  C's  are  $25  more 
than  one-third  of  A's  and  B's  together.  What  are  the  earnings 
of  each  ? 

21.  What  are  the  two  numbers  whose  sum  is  37  and  whose 
difference  is  23  ? 

What  problem  gives  a  formula  for  this  ? 


90  EQUATIONS. 

22.  A  boy  had  three  times  as  many  apples  as  oranges  ;  he 
sells  50  apples  and  15  oranges,  and  then  has  left  twice  as  many 
apples  as  oranges.     How  many  of  each  had  he  at  first? 

23.  Find  a  formula  for  problems  like  the  last. 

24.  A  boy  had  in  times  as  many  apples  as  oranges.  He 
sold  the  same  number  of  each,  and  then  had  n  times  as  many 
oranges  as  apples.  Selling  again  the  same  number  of  each  as 
before,  he  had  a  apples  left.  How  many  did  he  sell,  and  how 
many  of  each  had  he  at  first  ? 

25.  A  man  has  four  casks.  The  capacity  of  the  second  is 
f  of  the  first,  the  third  £  of  the  second  and  ^  of  the  fourth, 
and  the  first  holds  15  quarts  more  than  the  third  and  fourth. 
How  many  quarts  does  each  hold  ? 

26.  Divide  166  into  5  parts,  such  that  the  first  shall  be 
5  less  than  3  times  the  second,  2  less  than  twice  the  third, 
5  less  than  5  times  the  fourth,  and  10  more  than  three  times 
the  fifth. 

27.  A  man  rowing  uniformly  at  the  rate  of  four  miles  an 
hour,  rows  down  stream  one  hour,  then  resting  he  floats  with 
the  current  \  hour.  He  then  rows  back  in  3  hours.  How 
rapid  is  the  current. 

28.  A  rows  4  and  B  3  miles  an  hour.  A  is  14  miles 
farther  up  stream  than  B,  and  they  row  towards  each  other  till 
they  meet,  4  miles  above  B"s  position.  How  rapid  is  the 
current? 

29.  A  besieged  garrison  had  bread  to  supply  each  man 
with  12  oz.  per  day  for  5  weeks,  but  at  the  end  of  one  week 
they  lost  in  a  sally  200  men,  and  the  bread  remaining  was 
found  sufficient  to  give  each  man  10  oz.  a  day  for  6  weeks. 
How  many  men  had  they  at  first  ? 

Note. — Several  of  the  above  problems  may  be  solved  by  using  one, 
or  more  than  one  unknown  quantity.  These  the  student  may  solve 
again  by  using  more  than  one  unknown  quantity,  after  completing  the 
subject  of  Simultaneous  Equations. 


CHAPTER    IX. 

SIMULTANEOUS     EQUATIONS     OF     THE 
FIRST     DEGREE. 

241.  In  equations  containing  two  or  more  unknoivn 
quantities,  as, 

x  +  y  =  5. 
x  —  y—  i, 

the  values  of  x  and  y  in  any  one  equation  are  indeterminate  ; 
for,  whatever  value  be  given  to  one  of  them,  a  value  can  be 
found  for  the  other  which  will  satisfy  the  equation. 

Thus,  in 

x  +  y  =  5, 
whatever  value  be  given  to  y,  as  a,  we  have  only  to  make 

x  =  5  —  a, 
and  the  equation  is  satisfied. 

So  in  x  —  y  =  I,  if  x  =  i  +  a  and  y  —  a,  the  equation  will  be 
satisfied,  a  being  any  quantity  whatever. 

242.  But  if  we  assume  that  both  equations  are  true  at  the 
same  time,  that  is,  for  the  same  values  of  x  and  y,  the  case  will 
be  different.  The  number  of  values  each  of  the  unknown 
quantities  can  have  will  then  be  limited,  and  they  may  all  be 
found. 

243.  When  two  or  more  such  equations  are  satisfied  at  the 
same  time,  they  are  called  Simultaneous. 

It  is  evident  that,  if  the  two  equations  express  conditions  obtained 
from  the  same  problem,  in  which  x  and  y  represent  the  same  quantities 
in  both,  the  values  of  x  and  y  which  will  satisfy  both  equations  will  also 
satisfy  the  conditions  of  the  problem. 


92  SIMULTANEOUS     EQUATIONS 

244.  If  the  equations  actually  represent  different  condi- 
tions, they  will  be  independent ;  that  is,  one  of  them  cannot 
in  any  way  be  transformed  so  as  to  produce  the  other. 

The  equations 

x  +  y  —  5,     and     2X  +  2y  =  io, 
express  the  same  condition  and  are  dependent  equations ;  but 

x  +  y  =  5,     and    x  —  y  =  i, 
express  different  conditions,  and  are  independent. 

245.  If  a  problem  requires  the  finding  of  several  unknown 
quantities,  the  solution  of  the  problem  will  require  as  many 
independent  equations  as  there  are  unknown  quant  Hits. 

The  problem  must  therefore  furnish  a  like  number  of 
different  conditions,  each  of  which  must  be  expressed  by  an 
equation. 

Note. — It  is  not  necessary  that  each  of  these  equations  should  con- 
tain all  the  unknown  quantities.  It  is  only  necessary  that  all  the 
equations  should  contain  them  all. 


ELIMINATION. 

246.  The  reduction  of  simultaneous  equations  is  performed 
by  a  process  called  Elimination, 

This  process  consists  in  combining  two  equations,  con- 
taining two  or  more  unknown  quantities,  in  such  manner 
that  one  of  the  unknown  quantities  shall  disappear  from  the 
resulting  equation. 

247.  There  are  four  methods  of  elimination: 

ist.  By  Subtraction. 

2d.  By  Comparison. 

3d.  By  Substitution. 

4th.  By  Division. 


OF     THE     FIRST     DEGREE. 


CASE      I, 


93 


248.  For  Elimination,  by  Subtraction  we  have 
the  following 

Rule. — Multiply  each  of  the  equations  by  that  quantity 
which  will  make  the  coefficients  of  the  unknown  quantity  to  be 
eliminated  the  same  in  both  equations,  and  subtract  one  equa- 
tion from  the  other. 


i.  Given 

2»  +  zy  =  i3. 

(I) 

535  —  2y  =    4, 

(*) 

to  find  the  value  of  a; 

and  y. 

BY   SUBTRACTION. 

(i)  x  5  gives 

ipa;  +  i$y  =  65. 

(3) 

(2)    X    2         " 

IO£  —     41/  =     8. 

(4) 

(3)  -(4)    " 

190  =  57- 

(5) 

(5)  +  19     " 

y  =  3- 

Substituting  in  (i), 

2X  +   9  =   13. 

.'.      2X  =     4, 

and            x  =    2. 
CASE      IT. 

249.  For  Elimination  by  Comparison  we  have 
the  following 

Kule. — Find  from  each  equation  the  value  of  the  unknown 
quantity  to  be  eliminated  in  terms  of  the  other  unknown  and 
known  quantities,  and  equate  these  values. 


BY 

COMPARISON. 

Finding  x  from  (1), 

x  _  13  -  iy 
2 

(3)' 

Finding  x  from  (2), 

x  -  4  +  2y 
5 

(4)' 

Equating  (3)'  and  (4)', 

13  - 
2 

2>y  _  4  +  iy 

5 

From  which, 

y  =  3,     as  above. 

94  SIMULTANEOUS     EQUATIONS 


CASE      III. 

250.   For  Elimination  by  Substitution  we  have 
the  following 

Kule. —  Find  from  one  of  the  equations  the  value  of  the 
unknown  quantity  to  be  eliminated  in  terms  of  the  other 
unknown  and  known  quantities,  and  substitute  that  value  for 
this  unknown  quantity  in  the  other  equation. 


BY   SUBSTITUTION. 

Finding  x  from  (i), 

c  _  13  -  3V  . 

2 

(3)" 

Substituting  in  (2), 

x3  —  3V 
5        2  31       V  =  4- 

(4)" 

Giving  as  before, 

y  =  3- 

CASE      IV. 

251.  For  Elimination  by  Division  we  have  the 

Kule. — I.  Clear  the  equations  effractions  and  transpose  all 
the  terms  of  each  to  one  member. 

II.  Proceed  as  if  to  find  the  greatest  common  divisor  of  the 
polynomials  thus  found,  and  when  a  remainder  is  obtained  from 
which  one  of  the  unknown  quantities  has  disappeared,  put  this 
remainder  equal  to  zero  for  the  equation  sought. 


BT  DIVISION. 

Transposing  (1),                 ix  +  3?/  —  13  =  0. 

(3)'" 

Transposing  12),                5.7'  —  21/  —    4  =  0. 

(4)'" 

Dividing  (4)'"  x  2  by  (3)"' 

ioz  -    4?/  —    8  1  2*  +  33/  —  13 

iox  +  isy  —  65      5 

—  19^  +  57    .     .     .     Remainder. 

Putting  remainder  =  0, 

-  193/  +  57  =  0 

*<iy  =  57 

And,  as  before,                                  y  =    3. 

OF     THE     FIRST     DEGREE.  95 

Notes. — i.  The  reason  why  this  remainder  equals  zero  is  that  the 
dividend  and  divisor  each  equals  zero  ;  hence  the  remainder  must  be  zero. 

2.  The  method  of  elimination  to  be  employed  in  any  case  should  be 
that  which  will  render  the  work  most  simple. 

3.  When  the  method  by  Subtraction  is  used,  the  coefficients  of  the 
quantity  to  be  eliminated  must  be  made  the  same  in  the  two  equations, 
not  only  in  value,  but  also  in  sign.  If  their  signs  be  unlike,  they  may 
be  made  alike  by  changing  all  the  signs  of  one  of  the  equations,  or  the 
equations  may  be  added  instead  of  changing  the  signs  and  subtracting. 

Keduce  the  following  equations  : 


2. 

5x  +  3//  =  19, 

7X  —  21J   =      8. 

3- 

X         11 

-  +  -=«, 

2       3 

X         11 

6       9 

4- 

x       11 

-  +  |  =  m, 

a       0 

x       11 

=  n. 

c       a 

5- 

53  —  ly  =  —  8, 

5!/  +  3->  =  lx- 

6. 

as  -f  by  =  h, 

x  +  y    —  d. 

7- 

ax  +  bx  =  y~, 

x  =  hy. 

8. 

4X       2// 

7~y  =  4' 

6x  =  gy. 

9- 

x  +  y 

—   24, 
3 

4x  +  sy  =  Ii6- 

10. 

\x  +  toy  =  124, 

2£  +      9^/   =    I24. 

11. 

x  —  2ij  =  a, 

2x  +  Sy  =  b. 

12. 

ax  +  by  —  c  =  o, 

«'#  +  6'?/  —  c'  =  0. 

*3- 

^^  —  3.?y  =  to^/' 

4»  +  iy  =  //• 

14. 

y  =  ax  -\-  b, 

y  =  ax  +  &'. 

15- 

y  —  a  =  2  (x  —  b), 

y  —  b  =  2  (x  —  a). 

16. 

x  +  VJ  _  7l 
2        -  7*' 

4«  +  5?/  _  s 

4 

17- 

ft  +  3//  _  c 

3          _  3' 

£  —  3JJ        d 

3          "3 

18. 

2*  +  y 

4        =4, 

3*  —  3y 

6         -  Io 

a-  +  1        1 

19. 

_ 

2/           3 

a;            1 

y  +  1 


•  96  SIMULTANEOUS     EQUATIONS 

PROBLEMS. 

i.  A  number  consists  of  3  digits.  The  middle  digit  is  the 
sum  of  the  other  two,  the  first  "is  twice  the  lust,  and  inverting 
the  order  of  the  digits  gives  a  number  33  more  than  half  the 
first  number.     What  is  the  number? 

2.  A  person  spends  50  cents  for  apples  and  oranges,  buy- 
ing oranges  at  5  cents  and  apples  for  2  cents  apiece.  If  he 
had  taken  half  as  many  of  each  and  paid  6  cents  apiece  for 
oranges  and  one  cent  for  apples,  they  would  have  cost  him 
23  cents.     How  many  of  each  did  he  buy  ? 

3.  What  fraction  is  that  whose  numerator  being  increased 
by  5  and  the  denominator  decreased  by  2,  equals  1  ;  but 
whose  denominator  being  increased  by  5  and  the  numerator 
decreased  by  4,  becomes  |  ? 

4.  Three  pipes  discharge  into  the  same  cistern.  The  first 
and  second  will  fill  it  in  -\  hours,  the  second  and  third  in 
12  hours,  and  the  first  and  third  in  8^2  hours.  In  what  time 
will  each  pipe  fill  the  cistern  ? 

5.  A  certain  sum  of  money  at  interest  amounted  to  $550 
in  10  months,  and  to  8560  in  12  months.  What  was  the  sum 
and  the  rate  per  cent  ? 

6.  Two  persons,  A  and  B,  can  together  reap  a  field  of  grain 
in  10  days.  They  work  together  6  days,  when  A  is  left  to 
finish  the  work,  which  he  does  in  10  days  more.  In  v.  hat 
time  can  each  reap  the  field  ? 

7.  A  and  B  engage  to  do  a  piece  of  work  in  12  days,  but 
after  a  time,  finding  themselves  unable  to  accomplish  it,  C 
was  called  in  to  help  them,  and  the  work  was  finished  in 
time.  The  rate  of  working  of  each  was  such  that  A  could  do 
the  work  alone  in  f  the  time  required  for  B  to  do  it.  and 
C  could  do  it  with  A  in  f  of  the  time  in  wdiich  he  could  do  it 
with  B,  and  the  three  together  could  do  it  in  9  days.  What 
part  of  the  work  did  each  one  do  ?     How  long  did  C  work  ? 

8.  A  banker  has  two  kinds  of  money.  It  takes  a  pieces 
of  one  and  b  pieces  of  the  other  to  make  a  dollar.  If  c  pieces 
be  given  for  a  dollar,  how  manv  of  each  will  be  used  ? 


OF     THE     FIRST     DEGREE.  97 

9.  A,  B,  and  C  lunch  together.  A  furnishes  3  loaves  and 
B  2  loaves  and  a  basket  of  fruit,  the  whole  cost  of  which  was 
50  cents ;  but  0,  having  no  provisions,  agrees  to  pay  for  his 
share  25  cents  in  money,  when  it  is  found  that  only  2  cents 
of  this  will  belong  to  A.  What  was  the  cost  of  the  loaves  and 
what  of  the  fruit  ? 

10.  What  fraction  is  that,  whose  numerator  being  doubled 
and  denominator  increased  by  7,  the  value  becomes  f ;  but 
the  denominator  being  doubled  and  the  numerator  increased 
by  2,  the  value  becomes  f  ? 

11.  A  merchant  has  two  kinds  of  wine.  If  he  mixes 
a  gallons  of  the  first  with  b  gallons  of  the  second  the  mixture 
is  worth  c  dollars  a  gallon  ;  but  if  he  mixes  m  gallons  of  the 
first  with  n  gallons  of  the  second  the  mixture  is  worth  p  dol- 
lars a  gallon.     What  is  the  price  of  each  kind  of  wine  ? 

12.  What  two  fractions  have  their  sum  if,  and  the  sum  of 
their  numerators  equal  to  the  sum  of  their  denominators  ? 

13.  A  loaned  6500  in  two  separate  sums,  the  less  at  2% 
more  than  the  other.  If  the  per  cent  on  the  greater  be 
increased  and  that  of  the  less  diminished  by  1,  the  whole 
interest  will  be  increased  25$  ;  but  if  the  per  cent  on  the 
greater  be  so  increased  without  changing  the  other,  the 
interest  will  be  increased  2>2>i%-  What  were  the  sums  and 
the  rate  per  cent  of  each  ? 

14.  What  is  the  fraction  which  becomes  |  when  1  is  added 
to  the  numerator,  and  }  when  1  is  added  to  the  denominator  ? 

15.  Two  men  commence  business  at  the  same  time,  A  hav- 
ing $1000  more  capital  than  B.  At  the  end  of  a  year  A  had 
lost  an  amount  equal  to  B's  capital  and  B  had  gained,  the 
same  amount,  when  A's  capital  is  found  to  be  1 2 000  more 
than  B's.     What  was  the  capital  of  each  ? 

16.  Two  men  buy  a  farm  in  company  for  $2000,  each  put- 
ting in  all  the  money  he  had  and  giving  a  mortgage  for  the 
balance.  >{f  A  should  pay  the  mortgage,  he  would  then  have 
invested  §200  more  than  twice  as  much  as  B.  If  B  should 
pay  the  mortgage,  he  would  have  invested  $200  more  than 
A.  How  much  cash  did  each  put  in,  and  how  much  was  the 
mortgage  ? 


08  SIMULTANEOUS     EQUATIONS 


THREE     OR    MORE     UNKNOWN     QUANTITIES. 

252.  When  there  are  three  or  more  unknown  quantities, 
and  a  like  number  of  equations,  the  reduction  is  made  by  the 
following 

Eule. — I.  Eliminate  the  same  unknown  quantity  from 
different  pairs  of  the  given  equations,  thus  forming  a  set  of 
equations  independent  of  this  unknown  quantity,  and  one  less 
in  number  than  the  given  equations. 

II.  From  these,  in  like  manner,  eliminate  another  unknown 
quantity,  and  so  continue  till  an  equation  is  found  with  hut 
our  unknown  quantity. 

III.  From  this  find  the  value  of  that  unknown  quantity, 
and.  substitute  it  in  a  previous  equation  to  find  the  value  of 
another  unknown  quantity.  Substitute  these  two  to  find  a 
third,  and  so  on  till  all  are  found. 

The  following  example  will  illustrate  this  process: 

i.  Given           x  +  2ij  —  $w  +    %  =       4  (1) 

2X  —    y  +  2w  —  sz  =       1  (2) 

5X  —  ZV  —    w  —  2Z  =     11  (3) 

ix  +  41/  —  5W  +  6z  =  —  9  (4) 

to  find  the  value  of  x,  y,  w,  and  z. 

3x  +  by  —  qic  +  32  —  12  (5) 

5*  +  sy  -  I"  =  13  (6) 

(7) 
(8) 

(9) 
(10) 

(") 

(12) 

(13) 

443  +     gy  =     -  62  (14) 

44'1  -  88.y  =        132  (15) 

97y  =  -  104  (16) 

y  =  —     2 


(I)     X 

3. 

(2)   + 

(5), 

(I)    X 

0 

(3)  + 

(7), 

(I)     X 

6, 

(9)" 

(4), 

(8)- 

(6), 

(6)  x 

13. 

(10)  x   7, 

(12).- 

-  (13). 

(II)    - 

<  22, 

(14)" 

-  (15), 

(16)- 

-  97, 

2X  +  4y  —  bw 

+ 

23 

=    8 

"jx  +    y  —  "jw 

=  19 

bx  +  i2y  —  1S10 

+ 

bz 

=    24 

3.r  +     8y  —  1310 

=    33 

2X  —    \y 

=      6 

65.r  +  657/  —  giw 

=  169 

2\x  +  567/  —  917c 

-  231 

OF     THE     FIRST     DEGREE.  99 

Substituting  in  (n),  2X  +  8  =  6 

And  £C  =  —  X 

Substituting  in  (8),         —  7  —  2  —  7«o  =  19 

And  w  =  —  4 

Subst.  in  (1),             —1  —  4  +  12  +  3  =  4 

And  2  =  —  3 

.-.     x  =  —  1,    y  —  —  2,  w  =  —  4,     and    2  =  —  3,     ^.ns. 

2»  +  3#  —  42  =  8,  6.     5X—  iy+   z-\-   u  =  2, 

Z%  —  42/  +  2z  =  3,  ix— sy— $z+2u  =  2, 

4X  —  21)  —  32  —  5.  w— 2\j                   =  2, 

ax  +  by  +  cz  =  m,  x  +  5l/— 2Z          =  2. 

bx  +  cy  -{-  az  =  n,  1       £ 1 

ex-  +  ay  +  fo:  ==  r.  x      y  ~  a' 

x  +  y  =  12,  I    ,    T  _  I 

y-*  =    3,  y      z       y 

z  +  u=    7,  1       1  _  1 

W  +  £   =      8.  Z          £           C 

x      y      z  __  8.     aa    4-  Jy    +  «    =  o, 

2  ~  3        4  ~  «'«    +  #';/    +  c'2    =  o, 

»  _  y  ,  «  _  23 


«".T  +   Z»"?/   +   C"^    =   O. 

3       4       2        12  9.     «#  4-  bx  —  cy  =  m, 


n. 


%  ,  y -  =  -•  av  +  by  —  ex  = 

4     2     3 —  3 

10.     jb^2  =  a  (xy  +  y«  —  xz)  =  d  (xy  +  xz  —  yz) 

=  'c  (xz  -\-  yz  —  xy). 

PROBLEMS. 

i.  A  number  consists  of  4  digits.  The  first  is  half  the 
second;  the  third,  twice  the  second  plus  the  first ;  the  fourth, 
the  sum  of  the  second  and  third;  and  the  sum  of  the  digits  is 
15.     What  is  the  number? 

2.  Find  three  numbers,  such  that  \  the  sum  of  the  first 
and  second  shall  be  50,  \  of  the  second  and  third  shall  be  65, 
and  I  of  the  first  and  third  shall  be  55. 

3.  The  average  age  of  A,  B,  and  C  is  a.  The  average  age 
of  A  and  B  is  b,  and  of  B  and  C  is  c.    What  are  their  ages  ? 


100  SIMULTANEOUS     EQUATION'S. 

4.  Divide  the  number  150  into  three  parts,  such  that  \  the 
first  shall  be  \  of  the  second,  and  \  of  the  second  shall  be  \  of 
the  third. 

5.  A  person  has  2  horses  and  2  saddles,  all  of  which  are 
worth  £265.  The  poorer  horse  and  better  saddle  are  worth 
85  less  than  the  better  horse  and  poorer  saddle,  while  the 
better  horse  and  better  saddle  are  worth  845  more  than  the 
poorer  horse  and  poorer  saddle,  and  the  horses  are  worth 
5  times  as  much  as  the  saddles.  "What  is  the  value  of  each 
horse  and  saddle  ? 

6.  Three  brothers,  A,  B,  and  C,  bought  a  farm  for  -Si 000, 
A's  money  with  \  of  B's  and  \  of  C's  would  pay  for  it ;  so 
also  would  B's  money  with  -^  of  C's  and  ^  of  A's,  or  C's  with 
I  of  B's  and  \  of  A's.     How  much  money  had  each  ? 

7.  A,  B,  and  C  bought  a  farm  for  a  dollars.     A's  money 

with  —  of  B's  and  -  of  C's,  or  B's  money  with  — =  of  A's  and 
m  n  m 

—  of  C's,  or  C's  money  with  — 77  of  A's  and  —77  of  B's  will  pay 

iv  lit  ih 

for  the  farm.     How  much  money  had  each  ? 

8.  A  and  B  can  do  a  certain  piece  of  work  in  a  days : 
B  and  C  in  b  days;  C  and  D  in  c  days  ;  and  A  and  C  in 
cl  days.  In  what  time  can  each  do  the  work  alone,  and  how 
long  would  they  be  in  doing  it,  working  all  together? 

9.  Three  men  began  trade  at  the  same  time.  A  had  $2000 
more  than  twice  as  much  capital  as  B,  and  C  had  8500  less 
than  A  and  B  together.  The  first  year  A  gained  as  much  as 
B's  original  capital,  B  gained  as  much  as  A's  capital,  and  C 
gained  as  mnch  as  A's  and  B's  capital  together,  when  each 
had  an  ecpial  sum.     How  much  had  each  at  first? 

10.  A  doctor  visits  a  patient,  and  when  half  way  home  he 
is  overtaken  by  a  messenger,  and  called  to  return  3^  miles  to 
visit  a  second  patient ;  and  again,  when  half  way  home,  he  is 
called  to  return  to  visit  a  third  patient,  3  miles  farther  away 
than  the  first.  On  his  return  to  bis  office,  he  finds  he  has 
driven  20  miles.  How  far  did  each  patient  live  from  his 
office? 


CHAPTER    X. 

POWERS     AND     ROOTS. 

253.  A  JPowev  is  the  product  of  any  number  of  the 
equal  factors  of  a  quantity. 

254.  Powers  are  expressed  by  exponents,  which  show 
what  equal  factors  are  taken.     Hence, 

A  quantity  with  any  exponent  is  a  power. 

255.  The  quantity  itself  is  called  the  Base  of  the  power. 

256.  The  word  power  is  used  with  reference  to  the  effect 
of  a  quantity  as  a  Factor,  while  its  effect  as  a  Term  is 
called  its  value  or  magnitude. 

Thus,  if  I  of  the  factors  of  a  are  required,  it  is  written  a* ;  if  two- 
thirds  of  a  as  a  term,  it  is  written  \a. 

257.  The  operation  of  finding  any  part  of  the  factors  of 
a  quantity  is  similar  to  that  of  finding  any  part  of  its  terms. 

For  example,  f  of  the  terms  of  27  are  found  by  separating  27  into 
three  equal  terms  (27-7-3  =  9),  and  combining  two  of  them  (9  +  9  =  18,  or 

9x2  =  18). 

So  f  of  the  factors  of  27  are  found  by  separating  27  into  three  equal 
factors  (271  =  3),  and  combining  two  of  them  (3x3=:  9). 

258.  Evolution  is  the  process  by  which  a  quantity  is 
separated  into  any  number  of  equal  factors. 

259.  Involution  is  the  process  of  finding  the  product 
of  equal  factors. 


102  POWERS     A  X  D     ROOTS. 

.... 

XorE3. — i.  If  the  exponent  of  the  required  power  have  i  for  its 
denominator  (that  is,  if  the  exponent  be  integral),  the  factor  to  be  found 
by  evolution  will  be  the  base,  and  the  work  of  finding  the  required 
power  will  be  wholly  involution. 

2.  If  the  exponent  have  x  for  a  numerator,  the  work  will  be  wholly 
evolution. 

3.  If  both  numerator  and  denominator  be  1  (that  is,  if  the  exponent 
be  unity),  the  power  is  already  found  in  the  quantity  itself ;  that  is, 

260.  The  First  Power  is  the  quantity  itself,  or  base. 

261.  As  a  coefficient  shows  by  its  denominator  the  number 
of  equal  terms  into  which  a  quantity  is  to  be  separated, 
and  by  its  numerator  the  number  of  these  terms  that  are  to  be 
taken,  so  an  exponent  shows  by  its  denominator  the  number  of 
equal  factors  into  which  a  quantity  is  to  be  separated,  and 
by  its  numerator  how  many  of  these  factors  are  to  be  taken. 

262.  A  Hoot  is  one  of  the  equal  factors  of  a  quantity; 

as,  a*,  a*,  a?,  etc..  which  may  be  read,  "a  one-half  power, 

]  a  one-third  power, ,"  etc.;  or,  "the  square  {or  second)  root  of  a, 

flu'  cube   {or  third)   root  of  a,  the  fourth  root  of  a,""  etc. 

(Art.  58,  59.) 

263.  Quantities  with  fractional  exponents  are  called 
Bad  leal  Qua  tit  ities. 

They  may  be  expressed  by  the  radical  sign,  the  numerator 
of  the  exponent  remaining  as  an  exponent  of  the  base,  and 
the  denominator  being  placed  over  the  sign,  except  when  it  is 
two  (2),  in  which  case  it  is  omitted. 

Thus,    a*  =  <\/a ;    a%  =  tya- ;     a*  —  \/a3,    etc. 

264.  A  Power  which  can  be  expressed  without  a  fractional 
exponent,  as  a?,  is  called  Rational,  and  may  be  written  a2. 

When  it  cannot  be  so  expressed,  it  is  called  Irrational, 
or  Sard ;  as,  aK  a*,  etc. 

Thus,  8^,  4*,  and  8'  are  rational ;  but  S*,  $,  and  g3  are  irrational. 


POWERS     OF     MONOMIALS.  103 

265.   To   make   the   use   of    exponents    familiar   to    the 
student,  let  the  teacher  ask  such  questions  as  the  following: 

i.   What  is  the  value  of  16&?  16^?  i6*  ? 

2.  What  is  the  value  of  1 6*  ?  16*  ?  i6°? 

3.  What    is    the    value    of  16-1  ?  i6~2?     162  ?     i6~$  ? 
16-*?     1 6-°? 

4.  What  is   the   value    of    (16-1)2?     (16-1)-1?     (16  !)-2  ? 
325?    32-5  p    etc,  etc. 


POWERS     OF     MONOMIALS. 

266.  Any  power  of  a  quantity  may  be  expressed  by 
giving  it  the  exponent  of  the  required  power;  as, 

(8a362)t,        (cM  +  2£2)f. 

In  the  case  of  the  monomial,  the  parenthesis  may  be 
removed  by  applying  the  exponent  to  each  factor  separately. 

Thus,     (8a36-)i  =  8*  (azf  (62)1  =  4«2&*.     (Arts.  117,  118.)     Hence, 

To  find  any  power  of  a  monomial  we  have  the  following 

Eule. — Multiply  the  exponent  of  each  factor  of  the  mono- 
mial by  the  exponent  of  the  required  power. 

Notes. — 1.  It  will  be  observed  that  the  numerical  factor  or  coefficient 
must  be  included,  and  that  when  this  becomes  rational  the  required 
power  may  be  taken,  as  in  the  example  above. 

2.  This  rule  applies  to  all  powers  of  a  monomial,  integral  or  frac- 
tional, and  includes  the  work  of  evolution  and  involution. 

5.  Find  the  mth  power  of  (—  asb2). 

6.  Find  the  third  power  of  (x  —  y)*. 

267.  When  the  quantity  whose  power  is  taken  is  regarded 
as  having  a  directive  sign,  the  same  rule  applies  to  the  sign. 

Thus,  (—  aW)s,  which  equals  —  *  a?bk 

We  have  already  seen  the  force  of  the  signs  with  integral 
exponents.  Their  meaning  with  fractional  exponents  will  be 
considered  hereafter. 


104  BINOMIAL     FORMULA 


POWERS     OF     BINOMIALS. 

268.  Any  power  of  a  binomial  may  be  found  by  the 
Binomial  Formula,  given  below,  in  which  a  and  x 
represent  the  terms  of  the  binomial  and  n  the  exponent  of  the 
power,  which  may  be  integral  or  fractional,  positive  or  negative. 

Note. — We  are  not  yet  prepared  to  demonstrate  this  formula,  but 
the  student  may  verify  it  for  any  integral  powers  of  (a  +  x)  by  actual 
multiplication,  and  by  committing  it  to  memory  it  can  be  used  with  equal 
facility  before  or  after  demonstration. 

BINOMIAL     FORMULA.* 

(a  +  z)n  =  an  +  nan~xx  ^ s—    — '-  an~2x2 

x  '  1-2 

n  (n  —  i)  (n  —  2) 

H —    — *-* -a"-3*?,  etc. 

1-2-3 

269.  An  inspection  of  the  formula  will  show  that, 

I.  The  exponents  of  the  leading  letter  (a),  beginning  with 
the  exponent  of  the  power  (n),  decrease  by  unity  in  the  succes- 
sive terms. 

II.  The  exponents  of  the  follovring  letter  (x),  beginning 
with  o,  increase  in  like  manner  by  unity. 

III.  The  sum  of  the  exponents  in  any  term  equals  the 
exponent  of  the  required  power. 

IV.  Hie  coefficient  of  any  term  {after  the  first,  whose 
coefficient  is  1)  is  the  product  of  the  coefficient  of  tin-  preceding 
term  by  the  exponent  of  the  leading  letter  in  that  term,  divided 
by  the  exponent  of  the  following  letter  in  the  term  itself 

V.  The  number  of  terms  in  the  series  /rill  be  n  +  i  when  the 
exponent  of  the  poiver  is  integral  and  positive,  and  infinite  in 
all  other  cases. 

VI.  lite  mth  or  general  term  is 

11  (n  -  1)  (n  -  2)  ...  .  (w-ffi  +  a^^ 
1  -  2  •  3  •  4  .  .  .  .  (m  —  1 ) 
Note. — This  formula  translated  into  common  language  is  called  the 
Binomial  Theorem. 

*  This  formula  was  discovered  by  Sii  Isaac  Newton.    (See  p  30c.  Note  1.) 


INVOLUTION".  105 

270.  Although  the  terms  of  the  formula  are  all  +,  it 
must  be  remembered  that  as  a,  x,  and  n  may  oue  or  more  of 
them  be  negative,  the  sign  of  each  term  must  be  determined 
by  the  general  Rule  of  Signs.     (Art.  91.) 

271.  Signs. — I.  If  both  terms  of  the  binomial  are  positive, 
and  the  exponent  integral  and  positive,  the  terms  of  the  series 
will  all  be  positive. 

II.  If  both  terms  of  the  binomial  are  positive  and  the 
exponent  negative  {whether  integral  or  fractional),  the  terms  of 
the  series  will  be  alternately  positive  and  negative. 

III.  If  both  terms  of  the  binomial  are  positive,  and  the 
exponent  is  a  positive  fraction,  the  terms  of  the  series  ivill  be 
positive  until  the  term  whose  number  is  the  next  integer  greater 
than  (n  -f-  2).  This  term  will  be  negative,  and  the  following 
terms  will  be  alternately  positive  and  negative. 

IV.  If  the  second  term  of  the  binomial  be  negative  {the  first 
being  positive),  the  alternate  terms,  beginning  with  the  second, 
will  have  signs  opposite  those  given  in  the  cases  above. 

Note. — Let  the  student  verify  each  of  the  statements  in  Arts.  269 
and  271  by  an  examination  of  the  formula. 

INVOLUTION     OF     POLYNOMIALS. 

272.  In  finding  the  powers  of  polynomials,  we  must  per- 
form the  involution  and  evolution  separately.  The  square  of 
a  polynomial  may  be  written  by  Theorem  IV,  Art.  156. 

273.  The  cube  of  a  polynomial  may  be  written  by  the 
following 

Rule. — Tlie  cube  of  a  polynomial  is  equal  to  the  sum  of  the 
cubes  of  its  several  terms,  plus  three  times  the  products  of  the 
square  of  each  term  by  each  of  the  other  terms,  plus  six  limes 
the  products  of  the  terms  taken  three  at  a  time. 

Note. — Higher  integral  powers  of  polynomials  can  be  found  by 
actual  multiplication. 

274.  A  Multinomial  Theorem,  by  which  any  power 
of  a  polynomial  may  be  written  in  the  same  manner  as  powers 
of  a  binomial  by  the  Binomial  Formula,  is  sometimes  given, 
but  it  is  of  little  use  in  ordinary  mathematical  operations. 


106  INVOLUTION. 

EXAMPLES. 

Develop  the  following  by  the  Binomial  Formula  : 

i.     («  +  J).  =  rf  +  5^  +  5^*  +  L4J^ 

|£  |3 

|4  1 5 

=  ft5  +  5«4i  +  io«%2  +  loarb3  +  s«i4  +  J5,  Jms. 

2.  (ft  —  #)3  =  ft3  —  y&b  +  3«i2  —  bs,  A  us. 

3.  (a  +  5)4  =  a*  +  ia-U  +  iiipil  ar*P  +  etc. 

r 

=  a*  +  — ; i i  +  etc.,  ^««. 

4.  (ft  +  b  —  c  +  ft7)2  =  ft2  -+-  2a5  —  2ac  +  2ad  +  b-—2bc 

-\-  zbd  +  (?  —  zed  +  ft12,  -<4rcs. 

5.  (ft  +  #)2    and     (a  —  xf. 

6.  (2«  +  J)2     and     (ft  —  2b)2. 

7.  (a  +  by    and     (a  —  by. 

8.  (.r  +  y)6    and     (x  —  y)"'. 

9.  (2ft  +  2#)5     and     (3ft  —  3&)3. 

10.  («  +  a;)-1  and    (a  —  x)~z. 

11.  (ft  -f  «)*     and     (ft  —  x)z. 

12.  (ft  +  c)*    and     (ft  +  c)~?. 

13.  (ft  —  c)~^    and     (ft  —  c)*. 

14.  (ax  -f-  J^)*    and     (ax  —  by)~~. 

Expand  the  following  by  Theorem  IV  and  Rale,  Art.  156 : 

15.  (ft  +  zb  —  4c)2. 

16.  (2ft  —  3ft.r  +  lr  —  xy  +  zf. 

17.  (ax  +  by  —  32  +  5)2. 

18.  (ft  +  b  —  c  +  df. 

19.  (zx  —  3!/  +  s)3- 

20.  (ax  +  by  +  z  -\-  m  —  ay. 


EVOLUTION.  107 


EVOLUTION     OF     POLYNOMIALS. 

275.  To  find  any  Hoot  of  a  Polynomial,  the  mono- 
mial factors  should  be  removed  and  their  roots  taken  as  factors 
of  the  required  root. 

Let  P  represent  any  polynomial  whose  nth  root  is  required,  and  from 
which  the  monomial  factors  have  been  removed,  and  let  x  represent  one 
term  and  y  the  algebraic  sum  of  the  remaining  terms  of  the  root.     Then 

P  =  (x  +  yy  =  x"  +  nx"-*  y  +  etc., 
in  which  y  may  be  a  polynomial.     (Art.  268.) 

Now  if  none  of  the  literal  factors  of  x  are  found  in  any  term  of  y, 
the  root  sought  may  be  found  at  once  by  taking  the  n'h  root  of  that  term 
of  P  represented  by  xn  for  one  term  of  the  root,  and  dividing  all  the  terms 
represented  by  nx"~l  y  by  nx""1  for  the  rest  of  the  root. 

But,  as  some  of  the  factors  of  x  may  be  found  in  one  or  more  terms 
of  y,  {x+y)"  may  have  similar  terms,  which  by  uniting  will  prevent 
finding  by  inspection  the  value  of  nx:i~sy.  And  yet,  since  there  will  be 
at  least  one  term  of  y  which  does  not  contain  x  (the  monomial  factors 
having  been  removed),  we  shall  always  be  able  to  find  at  least  two  terms 
of  the  root,  which  being  raised  to  the  nlh  power  and  subtracted  from  P, 
will  give  a  remainder  from  which  other  terms  of  nxa~x  y  can  be  found. 
A  repetition  of  this  process  will  give  the  whole  root. 

276.  This  process  may  be  expressed  by  the  following 

Eule. — I.  Find  by  inspection  those  terms  of  P  which  are 
the  11th  powers  of  the  terms  of  the  root,  and  take  the  roots  of 
these  for  terms  of  the  root. 

II.  Representing  these  terms  by  x,  xx,  x2,  etc.,  find  the 
terms  representing  nxn'ly,  nxxn~x  y,  etc.,  and  divide  them 
respectively  by  nxn~1,  nxxn"x,  etc.,  for  other  terms  of  the  root. 

III.  Raise  the  part  of  the  root  so  found  to  the  nth  poiver, 
subtract  it  from  P,  and  find  in  the  remainder  other  terms  of 
the  farm  nxn~ly,  and  divide  as  before. 

Notes. — 1.  The  terms  to  be  found  by  Part  I  of  this  rule  will  be 
those  containing  the  highest  powers  of  the  different  literal  factors  ;  or,  if 
there  are  two  terms  that  contain  higher  powers  of  any  letter  than  any 
others,  the  one  containing  the  least  number  of  other  literal  factors  will 
be  the  term  required. 

2.  The  term.;  to  be  found  in  Part  II  will  be  those  containing  the  rie^t 
lower  powers  Qi  tl)  -  letters, 


108  EVOLUTION. 

277.  By  making  n  =  2,  we  have  the  rule  for  taking  the 
square  root,  and  n  =  3  gives  the  rule  for  the  cube  roof. 

278.  Any  Hoot  whose  index  is  the  product  of  two  or 
more  factors  may  be  found  by  taking  successively  the  roots 
indicated  by  those  factors. 

Thus,  the  sixth  root  is  the  cube  root  of  the  square  root. 

EXAMPLES. 

1.  Find  the  cube  root  of 

8a6  —  s6a5b  +  33a2bi  +  66a4b2  —  630:^  —  gab5  +  W. 

Solution. — By  Part  I  of  rule  we  find  that  8er6  and  ¥  must  be  cubes 
of  terms  of  the  root,  and  we  get  from  them  2a-  +  b*  as  a  part  of  the  root. 

By  Part  II  we  take  the  term  containing  the  next  lower  power  of  a, 
and  divide  it  by  n(2a2)a~1  =  3  (2a2)2  =  12a4  (u  in  this  case  being  3). 
This  gives  —  36a56  -f-  12a4  =  —  306. 

Also,  taking  the  term  containing  the  next  lower  power  of  b  and 
dividing,  we  have  —  gab5  -¥■  3&4  =  —  30b. 

By  each  of  these  last  two  steps  we  find  —  3«&  as  another  term  of  the 
root.  This  does  not  mean  that  —  yib  is  found  twice  in  the  root,  for  the 
same  term  of  the  root  will  frequently  be  found  more  than  once  by  the 
method  above. 

By  cubing  the  root,  2a'2  +  b'2  —  ytb,  we  shall  find  the  given  poly- 
nomial. 

2.  Find  the  cube  root  of  8»%  —  362;%*  -f  122^*  —  27.1- 
+  542%*  4-  272:%*  +  ifl  +  6.r%*  _  gxky  —  36./;7/;'. 

Solution. — From  Part  I  of  rule  we  have  2.rfyT  —  3a:*  +  y*  as  terms 
of  the  root,  and  Part  II  gives  no  other  terms.  This  root  cubed  gives  the 
polynomial. 

3.  Find  the  square  root  of  a2  —  2a?x  —  2azx  +  ^azx2—2ac 
+  a2x2  —  2as.c3  +  2acx-\-ah? —  2ah?-\-2azcx+ofa£ — 2a2cx2-\-c2. 

Solution. — By  Part  I  of  the  rule,  ±  c  is  one  term  of  the  root. 

By  Part  II,  dividing  the  terms  containing  c  first  power  (viz.,  —  2nd 
+  2acx  +  2(i'cx—2a*cx':)  by  ±2c  gives  the  otherterms,  to  ±  ax, 

;.    ±  (c  —  a  +  ax  +  aix  —  «'2.r°)  =  the  root  required. 


EVOLUTION-.  109 

EXAMPLES. 

Find  the  square  roots  of 

i   i 
i.     a  +  2a~x^  -f  x. 
i   i 

2.  a  —  2a~x~  +  x. 

3.  a%n  ±  2anxn  +  a*1. 

4.  a4  -f  4«3.f  +  6«2.c2  -f-  4«a;3  +  a^. 

5.  a6  +  6«5£  +  i5«4^  +  2ort3^3  +  i5«V  +  6ax5  +  x\ 

Find  also  the  cube  root  of  the  last  polynomial  and  of  the 
following  : 

6.  a6  +  3«4.£2  +  3«224  +  x6. 

7.  ^  +  |a  +  |aJ  +  1 

8.  «3  —  3«2a;  +  3ffi^'2  —  ic3. 

Find  the  fifth  root  of 

9.  a5  —  5a4  +  10a3  —  ioa2  -f-  5a  —  1. 

279.  The  square  root  of  numerical  binomials  of  the  form 
a  +  myb  may  often  be  found  by  separating  the  rational  term 
into  parts,  to  give  it  the  form  x2  ±  zxy  +  y2. 

10.  Find  the  square  root  of  7  +  4V3. 
Solution.        7  +  4\/3=4  +  4V/3  +  3  =  (2  +  V^)2- 

■"•     (7  +  4^/3)'  —  2  +  y^,  Ans. 

11.  Find  the  square  root  of  11  —  6V2. 

12.  Find  the  square  root  of  41  ±  12  V  5. 

13.  Find  the  square  root  of  33  ±  20V2. 

SIGNS      OF      ROOTS. 

280.  A  Power  of  a  quantity,  by  definition,  is  that  quan- 
tity affected  by  any  exponent  whatever;  while  a  Hoot  is 
the  quantity  affected  by  a  fractional  exponent  whose  numer- 
ator is  1. 

Note. — We  use  the  word  root  in  the  present  discussion  to  distin- 
guish such  powers,  and  not,  as  it  is  frequently  used,  for  fractional  powers 
in  general, 


110  EVOLUTION". 

281.  In  considering  what  sign  shall  be  given  to  fractional 
powers,  we  first  find  what  are  the  proper  signs  of  roots;  and 
as  all  fractional  powers  are  integral  powers  of  roots,  we  may 
find  their  signs  hy  Art.  91. 

For  example,  <n  is  the  third  power  of  the  fifth  root  of  a.  When  we 
know  the  sign  of  the  fifth  root  of  a,  the  third  power  of  that  sign,  or  that 
sign  taken  three  times,  will  be  the  sign  of  a>. 

282.  The  sign  of  a  root  is  found  by  the  general  rule, 
giving  the  sign  of  the  quantity  whose  root  is  taken  the  expo- 
nent of  the  root.     (Art.  90.)     Thus, 

( _  «2)i  _  _i a .         ( +  a2)i-  =  +ha,    etc. 

If  we  have  a  positive  quantity  (a),  it  may  he  written 
-f-  a,         — 2 a,         — 4  a,         —  6«,     etc.; 
or,  for  uniformity,  we  may  write, 

— ° a,  *         —2a,         — 4  a,         — 6 a,     etc.;     (1) 
using  any  even  power  of  —  to  express  the  positive  direction. 
So,  also,  if  a  be  negative,  it  may  be  written 

—  a,         —3a,         — 5  a,         — 7«,     etc.  (2) 

283.  If  now  we  take  any  root  of  +  a,  we  may  use  for  +  a 
any  of  the  expressions  in  (1) ;  or  if  we  take  a  root  of  —  a,  we 
may  use  any  of  the  expressions  in  (2).  This  will  give  for  the 
nth  root  of  -f-  «> 


— n  an,         — n  an, 

— *  an, 

—nan, 

etc. ; 

(3) 

and  for  the  nth  root  of  —  a, 

11               <    1 

— nan,             —"a". 

5    1 
— n  an, 

7    1 

etc., 

(4) 

n  being  any  integral  number. 

*  A^  o°,  which  has  no  power  as  a  factor,  equals  1,  ho  +,  which  hae  no  pov 
factor  of  direct  iou,  may  be  represented  by  — " , 


EVOLUTION.  Ill 

284.  From  this  we  see  that  we  may  have  different  signs 
for  the  same  root,  depending  on  the  sign  we  use  to  express  the 
direction  of  the  quantity  whose  root  is  taken.  These  are 
called  the  different  roots  of  a  quantity,  the  only  difference 
however  being  in  the  sign. 

285.  To  find  the  number  of  these  roots,  examine  first  the 
signs  of  the  roots  of  a  positive  quantity.     The  first  sign  in  (3)  is 

0 
_»  —  _o  _  _f_>      Hence, 

One  root,  of  whatever  degree,  of  a  positive  quantity  is 
positive. 

286.  The  other  exponents  of  the  signs  in  (3)  will  be  frac- 
tions until  we  come  to  one  whose  numerator  is  divisible  by  n. 
If  n  be  an  even  number,  we  shall  have  the  numerators,  giving 
a  root  whose  sign  is 

n 

— «    =    — l   =    — .       Hence, 

One  of  the  even  roots  of  a  positive  quantity  is  negative. 
If  n  be  odd,  the  second,  and  if  even,  the  third  integral 
exponent  of  —  will  be  —  =  2,  and  we  shall  have  the  root 


whose  sign  is  — 2  =  +  ,  which  is  the  same  as  the  first  root. 

Thee 

of  n, 


in 
The  exponents  of  the  sign  after  —  or  2  will  be,  for  all  values 


2  4 

2  +  -  ,        2  +  - ,        etc. 
n  n 

287.    As  it  will  not  affect  the   signs   to   drop  from  the 
exponent  an  even  number,  they  may  be  written 

~ ,         — ,        etc., 
n  n 

showing  that  the  series  will  repeat  itself  from  that  point,  and 
the  number  of  different  roots,  or  more  properly  of  different 
signs  for  the  root,  will  be  n. 


112  EVOLUTION. 

288.  If  we  now  examine  the  signs  of  the  roots  of  a  nega- 
tive quantity  as  given  in  (4),  we  shall  find  the  first  integral 
exponent,  when  n  is  an  odd  number,  to  be 

n 

giving  the  sign  —1=—.        Hence, 

One  of  the  odd  roots  of  a  negative  quantity  is  negative. 

289.  When  n  is  an  even  number,  we  shall  find  in  (4)  no 
integral  exponents  of  the  sign ;  but  whatever  be  the  value  of 
n,  we  shall  find  the  exponent 

271  4-   I  I 
—    =   2   +  -, 

11  n 

after  which  we  shall  get 

2  +  ^,         2  +  ^,        2+1-,        etc.; 
n  n  11 

and  as  we  may  omit  the  2,  the  exponents  repeat  themselves, 
and  we  have  as  before  the  number  of  roots  equal  to  n.  Hence, 
in  general, 

290.  Every  quantity  has  as  many  roots  as  there  are  units 
in  the  degree  of  the  root. 

This  may  be  illustrated  as  follows  : 

1.         (+  4)*  =  (_»4)*  =  (-44)^  =  (-64)",  etc., 

which  may  be  written 

+  2,  —  2,  — 2  2,  — 3  2,      etc., 

the  last  two  being  the  same  as  the  first  two,  and  we  have 
only  the  two  roots  +  2  and  —  2. 


2. 


(_4)i  =  (_s4)i  =  (_54)i  =  {-U)\  etc.;    or 


i 


_l  2,  —-2,  — *  2,     etc.; 


which  give  only  —^2  and  —*2,  the  others  being  the  same 
as  these. 


IMAGINARY     QUANTITIES. 


113 


3.  (+  8)3  =  (—2  8)3  =  (— 4  8)^  =  (— 6  8)3,  which  become 

+   2,  —  »   2,  —  i  2,  — 22, 

and  we  get  three  cube  roots  of  +  8. 

4.  (_S)i  ■  =  (-38)i  =  (-5S)i  =  (-78)^;    or, 

— ia,  —2,  -^2,  —^2  =  (— *a), 

and  we  have  also  three  cube  roots  of  —  8. 

In  the  same  way  the  student  may  find  4  fourth  roots  of  16, 
and  5  fifth  roots  of  32,  etc. 

291.  We  have  taken  rational  quantities  for  illustration, 
but  the  same  would  of  course  be  true  for  irrational  quantities. 

We  have  found  that  the  signs  of  some  of  these  roots  are  + 
and  some  — ,  while  most  of  them  can  only  be  expressed  by  the 
sign  —  with  a  fractional  exponent. 


IMAGINARY     QUANTITIES. 

292.  The  force  of  the  sign  — ,  with  any  integral  exponent, 
has  already  been  explained.  (Arts.  90,  91.)  It  now  remains 
to  consider  what  interpretation  shall  be  given  it  when  affected 
by  a  fractional  exponent. 

Let  AB,  Fig.  1,  be  a 
line  whose  length  is  2, 
and  whose  direction  is 
positive,  reckoned  from  A 
to  B.  Then  AC,  lying  in 
the  opposite  direction,  will 
=  —  2,  and  these  lines  will 
represent  the  sq.  roots  of 
+  4.  Since  the  sign  — , 
in  reversing  the  line  AB,  c 
turns  it  in  the  direction 
indicated  by  the  arrow 
(Art.  go),  when  it  has  ex- 
pended J  its  power,  the 
line  will  have  the  direction 
AD,  which  will  therefore 
be  expressed  by  —\  2.  In 
like  manner,  the  -I)  power  of 


4-  2 


1U 


I  M  A  G  1  :N'  A  It  Y      (iL'AXTITIES. 


this  sign  (— ')  brings  the  line  to  AE.     These  lines  represent  the  square 
roots  of  —  4. 

In  like  manner,  all  fractional  powers  of  —  indicate  directions  out  of 
the  line  +  and  — ,  which  are  found  by  taking  such  part  of  a  reversal  as 
the  exponent  of  the  power  indicates. 

293.  The  directions  of  the  cube  roots  of  a  positive  quantity 
are  represented  by  the  lines  AB,  AD,  and  AE,  Fig.  II. 


Fig.il. 


If  these  lines  be  taken  2  units  in  length,  they  will  represent  the  cube 
roots  of  +  8  both  in  magnitude  and  direction. 

So  also  AF,  AC,  and  AG  represent  the  cube  roots  of  —  8. 

294.  A  Ileal  Quantity  is  one  whose  sign  is  either 
+  or  — . 

295.  An  Imaginary  Quantity  is  one  whose  sign  is 
— ,  with  some  fractional  exponent.  • 

Note. — In  the  solution  of  problems,  quantities  are  always  considered 
as  lying  in  a  certain  line  in  one  of  two  opposite  directions,  called  positive 
and  negative.  (Art.  86,  3  .  >  Any  quantity  not  in  that  line  is  regarded  as 
unreal,  and  is  therefore  called  imaginary.  Such  a  quantity  in  a  result 
indicates  the  introduction  of  some  impossible  condition. 


IMAGINAKY     QUANTITIES.  115 

296.  From  (Arts.  281-289)  we  deduce  the  following 

PRINCIPLES. 

i°.  Of  the  even  roots  of  a  positive  quantity,  tivo  are  real, 
one  +  and  the  other  — . 

20.  Of  the  odd  roots  of  a  positive  quantity,  one  only  is  real, 
and  its  sign  is  + . 

30.  Of  the  odd  roots  of  a  negative  quantity,  one  only  is  real 
and  Us  sign  is  — . 

40.  All  the  even  roots  of  a  negative  quantity  are  imaginary. 

297.  In  mathematical  computations,  the  real  roots  of 
quantities  are  used  in  all  cases,  when  there  are  such ;  hence 
the  only  case  which  necessitates  the  use  of  imaginary  quantities 
is  that  which  requires  the  even  root  of  a  negative  quantity. 

For  this  reason,  an  imaginary  quantity  is  often  defined  as 
the  even  root  of  a  negative  quantity. 

298.  In  Fig.  II,  if  the  lines  DE  and  FGT  be  drawn  perpen- 
dicular to  BO,  then 

AJST  =  +i;  MD  =  (-3)*5 

AM=-i;  NO  =-(-3)*; 

NF  =  -M  =  (-  3)*;  ME  =  -  (-  3)l 

Note. — The  student  who  understands  Trigonometry  may  verify 
these  values. 

AN  +  NF  =  1  +  V^3; 
AN  +  NG  =  1  —  V^3 ; 
AM  +  MD  =  -  1  +  V^3  ; 
AM  +  ME  =  -  1  -  V~~2,. 

For  all  purposes  of  mathematical  calculation, 

AN  +  NF  =  AF ; 
AM  +  MD  =  AD ; 

AN  +  NG  =  AG ; 
AM  +  ME  =  AE. 


116  IMAGINAKY     QUANTITIES. 

299.  llence  we  have  the  following  equations: 

— *2  =  I  +  V-  3;  (0 

_§2  =  t  _  V— 3;  (2) 

-»2    =    -I    +    V—  3;  (3) 

_#  2   =  .—  I  —  V—  3-  (4) 

The  first  two  are  the  imaginary  cube  roots  of  —  8  and  the 

last  two  of  +  8. 

Notes. — 1.  It  is  evident  that  travelling  over  the  distance  AF  from  A  is 
equivalent  to  travelling  over  AN  and  NF  successively,  so  far  as  the  result 
is  concerned. 

2.  But  it  is  not  so  evident  that  the  second  members  of  these  equa- 
tions may  be  used  for  the  first  members  in  mathematical  computations. 
The  student  may,  however,  verify  the  statement.     For  example, 

1.  Add  the  1st  and  4th,  and  we  have,  0  =  0. 

2.  Add  the  2d  and  3d,  and       "      "  o  =  o. 

3.  Cube  the  1st  or  2d,  and       "      "  —  8  =  —  8. 

4.  Cube  the  3d  or  4th,  and      "      "  8  =  8. 

Observe  in  adding  that  —  *  2  and  — »  2  lie  in  opposite  directions,  and 
being  equal  in  distance,  their  sum  is  zero. 

300.  The  forms  in  the  second  members  of  these  equations 
are  always  used  instead  of  those  in  the  first  members.  The 
advantage  of  this  will  be  readily  seen  in  the  fact  that  the 
second  form  has  its  imaginary  part  always  a  square  roof,  which 
will  lie  along  a  line  at  right  angles  to  BC,  in  one  of  two 
opposite  directions. 

Such  quantities  may  therefore  be  added  and  subtracted  like 
real  quantities. 

Thus,  we  may  add 

-  1  +  V-~3 
and  —  1  —  y  —  3 
Sum,     —  2 

which  is  represented  by  the  line  AC     But  we  cannot  by  any  algebraic 
process  reduce  — '  2  and  —  *  2  to  one  term. 


IMAGINARY      QUANTITIES.  117 

Notes. — i.  We  may,  however,  see  that  by  a  different  method  of 
addition  (not  algebraic),  the  sum  of  —  •>  2  and  — j  2  is  —  2  ;  for  if  we  go 
from  A  to  D,  and  then  from  D  a  distance  and  direction  equal  to  —  *  2, 
that  is,  from  D  to  C,  we  shall  reach  the  same  point  C  to  which  the  line 
AC  or  —  2  extends. 

2.  To  find  the  imaginary  roots  of  a  number  in  this  binomial  form 
requires  a  knowledge  of  Trigonometry.  The  student  who  is  not 
acquainted  with  the  fundamental  principles  of  that  subject  will  not 
fully  understand  Arts.  301-305.  It  will,  however,  be  for  his  advantage 
to  read  them. 

301.  The  process  of  finding  imaginary  roots  is  the  same  as 
finding  the  base  and  ■perpendicular  of  a  right-angled  triangle, 
when  the  liypothenuse  and  angles  are  given. 

The  liypothenuse  is  the  real  root  of  the  quantity,  obtained 
in  the  nsnal  way  by  evolution,  and  the  angle  at  the  base  is 
found  by  the  sign  of  the  root. 

302.  Since  the  sign  —  represents  an  angle  of  1800,  we 
have  for  these  angles,  for  the  roots  of  positive  quantities, 

o°        3600        2-360°        3-360° 
n  '        n    '  n      '  n 

and  for  the  roots  of  negative  quantities, 

180°         viSo°        5-iSo°        7-180°  ..         0    , 
, ,      ^ , ,     etc.     (Art.  283.) 

Note. — Any  one  of  these  expressions  which  represents  a  multiple  of 
1800  gives  a  real  root.    The  rest  are  imaginary. 

Whatever  be  the  value  of  n,  the  angles  less  than  i8o°  will 
correspond  to  those  greater  than  i8o°.  That  is,  if  we  have  an 
angle  equal  to  180°  —  (3°,  there  will  be  another.  iSoc  -f-  (3°. 
Hence  the  roots  will  always  be  found  in  pairs  of  the  form 

a  ±  V^b. 

303.  The  two  roots  having  these  relations  to  each  other 
are  called  Conjugate  Hoots.  The  formula  for  these,  by 
Trigonometry,  will  be 

r  (cos  0  ±  V '—  sin2  0), 

in  which  r  is  the  real  root  of  the  quantity,  and  0  the  angle 
indicated  by  the  sign. 


etc.; 


118  IMAGINARY      QUANTITIES. 

EXAMPLES. 

i.  Find  the  imaginary  values  of  32^. 

Here        n  =  5,        0  =  72    and  144%         and        r  = 
By  the  formula  the  imaginary  roots  are 


2  (cos  720  ±  ^/—  sin2  72"), 
and        2  (cos  144  °  ±  ^  —  sin-  1440). 

Introducing  approximate  values  for  sine  and  cosine  of  72°  and  144' 
we  have 


2  (.309  ±  -V/-.9512), 
and        2  (—  .809  ±  yf—  .588s). 

2.  Find  the  imaginary  values  of  (—  32)^. 

Here        «.  =  5,         r  =  2,         and        0  =  36"  and  1080. 
Hence  the  imaginary  roots  are 


2  (cos  36.0  ±  \/—  sin-  36°), 
and        2  (cos  108°  ±  y/—  sin2  108°). 
Or,  substituting  values  of  sine  and  cosine  of  36"  and  108°, 

2  (.809  ±  -v/^Tsss2), 
and        2(—  .309  ±  y/ —  .9512). 

Note. — Hence  it  appears  that  the  imaginary  fifth  roo:s  of  32  and  of 
—  32  are  the  same,  except  in  the  sign  of  the  real  term. 

3.  Find  the  imaginary  values  of  (729)1'  and  ( —  729)*. 

For  the  first,         n  =  6,         r  =  3,         6  =  6o°  and  120°. 
The  values  are  therefore 


3  (cos  6o°  ±  y/ —  sin2  6o°), 

and        3  (cos  1200  ±  y/  —  sin-  1200). 

Or  3  (£  ±  hV^jl 

and  3  (-  1  ±  */v/~3). 

For  the  second,        n  =  6,        r  =  3,        0  =  300,  90°,  and  1500. 
The  values  are 

3(iV3±i\/:i2)'     3(°±\/-*)>    and     3(-iV3±*A/-2) 
The  second  pair  of  values  reduce  to  ±  —  *  3  =  —  *  3  or  —  *  3. 


IMAGINARY     QUANTITIES.  119 

304.  From  the  preceding  we  have  a  direct  method  of  find- 
ing  any    power  of    an    imaginary   expression   of    the    form 

a  ±  V—  b.  Since  a  is  the  base  and  \/b  the  perpendicular 
of  a  right-angled  triangle  whose  liypothenuse,  with  a  proper 
sign,  is  equivalent  to  the  binomial  imaginary,  we  may  operate 
by  the  following  rule,  in  which  v  represents  the  liypothenuse 
and  m  the  proper  exponent  of  —  to  give  the  quantity  the 
right  direction. 

Eule.  —  I.    Change  the  expression    to   the  form    — m  r. 
(Art.  305.) 

II.  Raise  the  resulting  monomial  to  the  required  power. 
(Art.  266.) 

III.  Restore  the  form  a  ±  V—  b.    (Arts.  302,  303.) 

305.  To  make  the  change  from   a  ±  V —  b  to  —  mr  \vt 
have  by  Trigonometry, 

r  =  \/a2  +  b; 

m  =  -7T-03  in  which  6  =z  cos-1  -• 
1 80  r 

The  following  examples  will  illustrate  the  process: 
1.  Find  the  cube  root  of  —  4  +  4  V—  3- 

SOLUTION. 


m  =  ——- 
180 

II. 

III. 

— »  2  = 

1.  r  =  \/d1  +  b  =  ^16  +  48  =  8  ; 
0     _  coa-'  (—  I)  _  120°  _  2 

-  180°         ~  180°  —  3" 

-4  +  4  \/—3  =  -l8. 

(-S8)»  =  -3.2 

2  (cos  40°  +  y^sin2  400) 
=  2  (.766  +  y/—  643?),  Ans. 
This  gives  one  oikthe  cube  roots  of  —4  +  4  /y/—  3.     The  other  twc 
will  be  found  by  using  for  —3  8,  its  equal  —5  8,  or  — "  8.     (Art.  290.) 

2.  Find  the  5th  root  of  243  (£  —  \  \/—  3). 

3.  Find  one  pair  of  the  imaginary  values  of  64*. 

4.  Find  the  cube  roots  of  —  125. 


120  RADICALS. 

CALCULUS     OF     RADICALS. 

306.  AH  mathematical  operations  upon  radicals  are  per- 
formed by  the  same  rules  as  like  operations  upon  rational 
quantities.  The  student  need  only  become  familiar  with  the 
use  of  the  signs  and  symbols  of  the  algebraic  language,  and 
with  the  principles  involved  in  the  fundamental  mathematical 
operations,  and  then  apply  them  alike  to  rational  and  radical, 
to  real  and  imaginary  quantities. 

Any  attempt  to  make  a  difference  in  their  application 
to  real  and  imaginary  quantities,  is  liable  to  result  in 
confusion.  One  principle  only  in  addition  to  those  already 
given  requires  attention,  viz., 

Quantities  wl/ose  signs  are  neither  the  same  nor  oppo- 
site, cannot  be  united  in  one  term. 

Thus,  2>a        aud         —  =  5a 

are  neither  the  same  nor  opposite  in  direction,  and  therefore  cannot  be 
united  in  the  same  term.  They  can  be  added  only  by  writing  them  with 
their  signs ;  thus, 

3«  — *  yi, 
or,  as  it  is  frequently  written, 

3a  +  5«V—  r> 
in  which  \/—  1  has  no  force  except  as  &  factor  of  direction,  1  having  no 
power  as  a  factor. 

Those,  however,  who  prefer  this  form,  can  use 


-y/—  1         for  y'- 

and        —  \/~  1        for    —  \/- 


or 


or 


1 


It  is  wholly  immaterial  which  form  is  used,  if  the  student 
does  not  allow  the  presence  of  the  1  to  give  him  the  impres- 
sion that  something  besides  a  direct  ire  sign  is  intended. 

307.  For  Addition  and  Subtraction  of  Radicals 

we  have  the  following 

Rule. —  Change  the  signs  of  terms  to  be  subtracted,  a  id 
unite  similar  real  terms  in  one;  also  similar  imaginary 
terms,  and  connect  dissimilar  terms  by  their  signs. 


CALCULUS     OF     RADICALS.  121 

Note. — Changing  signs  means  reversing  the  direction  by  applying  a 
minus  sign  to  the  term.  Thus,  changing  the  sign  —I  makes  it  —  i ;  that 
te,  +  y/~  becomes  —  y/ — ■ 

i.  Simplify  ab  —  V—  a  +  2\/ —  a  +  sab. 

Ans.  ^ab  +  V  —  a. 

2.  Subtract  2a  +  3V —  b  from  7a  —  5  V —  b. 

Ajis.  $a  —  8a/— #• 

3.  Subtract  —  *8  from  — *  12. 

Changing  sign  of  subtrahend,  it  becomes  —  *  8  and  therefore  lies  in 
the  same  direction  as  —  ^  12.  Adding,  after  changing  the  sign,  gives 
—  a  20,  J./2.S. 

308.  For  Multiplication  of  Radicals  we  have  the 
following 

Eule. — I.  ^o  tfZie  product  of  the  numerical  factors  annex 
the  literal  factors  each  with  an  exponent  equal  to  the  sum  of  its 
exponents  in  the  several  factors.     (Art.  127.) 

II.  Give  the  product  the  sign  — ,  with  an  exponent  found 
by  adding  the  exponents  of  —  in  the  several  factors,  and 
subtracting  from  the  turn  the  greatest  even  number.  (Art.  127.) 

4.  Multiply  a  +  y/ — b  by  a  —  y/ —b. 


OPERATION. 

a   +     y/—  b 

a   —    V '—  6 

a2  +  ay/—  6 
—  a\/—  b  +  b 

a2                +  b, 

Ans 

Note. — a-  is  positive,  because  every  power  of  +  is  + ,  or  because  in  both 
factors  the  sign  —  has  zero  for  an  exponent.  The  product  of  a  x  (4-  y/—b) 
has  the  sign  —  -  or  +  y/  —  ,  the  sum  of  the  exponents  of  —  being  \. 
Also,  a  x  (—  <y/—  b)  has  the  sign  —  -  or  —  y/ — ,  for  the  same  reason; 
and  ( +  y/  —  b)  x  (—  y  —  b)  has  the  sign  + ,  for  we  have  in  one  — •  and 
in  the  other  —  %  giving  —  '•'  or  +. 


122  CALCULUS     OF     RADICALS. 

309.  For  Division  of  Radicals  we  have  the  following 

Eule. — I.  Diciile  the  numerical  factor  of  the  dividend  by 

the  numerical  factor  of  the  divisor,  and  annex  the  letters  of 
both  dividend  and  divisor  each  with  an  exponent  found  by 
subtracting  its  exponent  in  the  divisor  from  its  exponent  in  the 
dividend. 

II.  Give  the  quotient  the  sign  — ,  with  an  exponent  equal  to 
its  exponent  in  the  divide/id  minus  its  exponent  in  the  divisor, 
adding  an  even  number  sufficient  to  make  this  exponent  positive. 

Note.— These  rules  apply  especially  to  monomials,  but  operations 
upon  polynomials  are  made  up  of  operations  on  monomials. 

5.  Divide  12  ( —  a"jP)^  by  —  3  (ab3)?. 

Operation. — Dividing  12  by  3  gives  the  numerical  coefficient  4,  to 
which  annex  the  letters;  thus,  $ab. 

For  the  exponent  of  a,  we  have  §  —  1-  =  1,  and  for  b,  ?,  —  |  =  i, 
both  of  which  we  omit. 

For  the  exponent  of  —  we  have  -J-— 1  =  —i,  to  which  add  2, giving  £. 
The  quotient  therefore  is  —  *  4^6. 

If  in  this  example  we  write  the  sign  without  changing  its  exponent, 
it  would  read  —  ~?  40b.  This  does  not  differ  from  —  5  $ab,  except  that 
the  former  supposes  the  revolution  to  be  made  in  the  opposite  direction, 
so  that  —  ~i  indicates  the  direction  of  a  line  (Fig.  1)  turned  downward 
from  AB  till  it  comes  to  AE  ;  while  —  *  indicates  the  same  direction 
produced  by  turning  AB  in  the  direction  indicated  by  the  arrows." 

310.  The  only  reason,  therefore,  why  we  add  to  or  subtract  from 
the  exponent  of  —  an  even  number,  is  that  by  so  doing  we  express  all 
signs  by  +,  or  by  — ,  with  an  exponent  greater  than  o  and  less  than  2, 
and  by  using  the  binomial  form  for  imaginary  roots,  we  have  all  quan- 
tities reduced  to  four  directions,  viz.,  +  ,  — ,  — «,  and  — !.  The  first  two 
are  real,  and  last  two  imaginary.     The  imaginary  may  be  written 

-y/—  and  —  \/— , 

or  -y/—  1  and  —  \/—  1. 

Perform  the  indicated  operations  in  the  following: 
6.     2«w  x  $a*xs  —■  6ax. 


7.  2\/ax  -f-  2>\/a~x  —  4V —  ax  +  2\/ —  ax. 

8.  Vf<2  —  x*  4-  2  Va2  —  x*  -  5  (fl2  —  a*)*  +  4  (tf2  —  *2)*- 

*  This  agrees  with  the  treneral  use  of  +  and  — ,  for  if  a  positive  exp  i«  nt  indicates 
revolution  in  one  direction,  a  negativt  expont  nt  Bhould  indicate  revolution  in  the  opp<x 
site  direction. 


CALCULUS     OF     RADICALS, 


123 


[(«  +  z)*  +  («  —  »)*]  x  [(a  +  s)1  —  (a  —  »)*]. 
a26*  +  a*62  —  ai*  +  ah)  x  (a*£  +  «£*). 
a  +  2a^-  +  J)  h-  (a*  +  fl£). 

a  —  S)  -v-  (a*  —  £s). 

za  _  52)  ^_  (ai  _  si). 

c$$  +  2  (a5)i  —  $a$ }  —  j  (a&)^  +  20$  \. 

—  Va  x  —  V—  b  x  V—  ex—  V^d. 

-  a  ±  aV—  3)3- 
±  V—  x)  {a  =F  V~x). 

2  +  a)  ~  (a  —  V~-^x). 
a  —  x)  -~-  (V—  a  +  V-- «)• 
a3  —  2a^  +  azx)i. 


a  +  2cfixz  +  a:. 

1    K     .    n   m 


■     lit     11  1      i\       ,      n     in  1       1  v 

a3»  _  J2»)  ^_  (a»   +  jn) 
fl3  _  J)   _u   (a  _  jfy 

a2  +  £3)  -S-  (<£  +  J). 
«3  _  J8)  _;_  (ai  _  h\y 

a*  +  J2)  ^-  («4  +  jf). 
a*  _  J8)  -=-  (a§  +  &*). 

(—  1  +  Vs  +  ^—  10  -  2  Vs)5. 


(-1  +  v/5-^/-io-2V/5)5. 
(-  1  -  V5  +  v '-  10  +  2V5)3. 
(—  1  —  \/5  —  'V//—  10  +  2V5)5. 


124  RADICALS. 

FORMER     NOTATION. 

311.  It  has  already  been  suggested  that  the  notation  —  * 
takes  the  place  of  the  more  common  notation  ( —  i)-,  or 
\/ —  i.  To  show  how  the  two  methods  compare,  we  give  the 
following  examples  in  Multiplication  and  Division  of  imagi- 
nary quantities. 

312.  By  the  Common  Method  of  notation  every 
imaginary  term  is  considered  as  the  product  of  two  factors, 
one  of  which  is  V —  i,  the  other  heing  a  real  quantify  with 
the  sign  +  or  — .  In  multiplying  or  dividing,  these  factors 
are  considered  separately.  It  is  necessary,  therefore,  to  find 
the  various  integral  poivers  of  V —  i. 

Since  the  square  of  any  square  root  is  the  quantity  itself, 
we  have 

(V^)1  =  V~ 


i 


V—  i  x  V—  i  =  {V—  i) 


2   — 


I 


I 


—  i  x  V—  i  =  {V—  03  =  —  V- 
—  V^i  x  V^  =  (V^O4  =  —  (—  i)  =  +  i ; 
i  x  V—  i  =  {V—  i)5  =  v '  —  i. 

It  is  evident  that  the  higher  powers  will  not  give  any  new 
forms,  and  we  have  four  forms  onlv,  viz. : 


V—  i ;     —  i  ;     —  V—  i ;    and     +  i  ; 
corresponding  to  the  explanation  (Art.  292)  and  the  illustra- 
tion (Fig.  1). 

EXAMPLES. 

1.  What  is  the  product  of  —  V —  a  by  —  V—  b  ? 
Solution. — Resolving  each  into  factors,  we  have, 

—  ^/—  a  =  —  ^/a  \/—  1 ; 

—  ^/—  b  =  —  \/b  \/—  1 ; 
—  *<Ja  x   —  *>Jb  =  -\/ab ; 

<y/—  I   x   a/-^i  =  —  i  ; 
—  1  x  -y/ad  =  —  *Jab. 


RADICALS.  125 

That  is,  we  resolve  each  quantity  into  two  factors,  one  of  which  is  real 
and  the  other  the  imaginary  expression  \/—  1,  then  multiply. 

Let  the  student  multiply  the  following,  using  either  nota- 
tion at  pleasure. 

2.  Multiply  V  —  4  by    —  V—  3« 

3.  Multiply  3  +  V—  2   by  3  —  V—  2. 

4.  Multiply  V—  9   by   V  —  16. 

5.  Multiply  V—  3   by   V—  24. 

313.  In  division  the  factor  V —  1,  found  in  both  dividend 
and  divisor,  will  cancel,  and  the  quotient  will  be  the  cpaotient 
of  the  real  factors. 

EXAMPLES. 

1.  Divide    V —  a   by    V —  b. 
Solution.  y'—  a  =  ^/a  ^  —  1 ; 

v7—  t>  =  Vb  V—  J ; 

\/a  V '—  1        \/a 

-    _       _—  =  -*-=.    Ans. 

y/b  -\/—  1       ^/b 

2.  Divide   v^  by    V —  J. 

Solution.     >y/a  =  —  -y/a  x  —  1  =  —  \/a  \/—  1  <\/—  1 ; 
y/—  b  =  \/b  \/—  1; 
—  's/a  -y/—  1  \/—  1  _  —  \/a  <\/— _£  _ 


^/b  *J  —  1  yft 


=  -| 


Ans. 


3.  Divide  9  v7—  8   by   3  v7 —  4- 

4.  Divide  4  V —  3   by   2  y —  9. 

5.  Divide  4V — «2   by    2a  V — 2. 

6.  Divide  4  -f-  v7 —  2   by   2  —  v —  2. 


126  REDUCTION. 

REDUCTION     OF     RADICALS. 

314.  Radical  expressions  may  frequently  be  simplified: 

ist.  By  removing  or  introducing  rational  factors. 

2d.    By  reducing  radicals  to  like  indices. 

3d.    By  rationalizing  one  term  of  a  radical  fraction. 

315.  To  Remove  Rational  Factors. 

Rule.  —  Separate  the  quantity  under  the  radical  sign 
into  rational  and  irrational  factorsx  and  taking  the  required 
root  of  the  rational  factors,  write  them  outside  the 
radical  sign. 

1.  Given  (2^a3bx2)^  +  (gaPbx2)*  —  (i6azbx2)?,  to  simplify 
the  expression. 

OPERATION. 

(25a3&E2)*  =  (25«2a;9.  atifi  =  {2$a?x°f  (ab)*  =  sax{ab)^ ; 
(gasbx^  =  (g^x2)1  (ab)i  =  ?,ax  {abf 
(i6a3bx-)--  =  4ax(ab)?. 
.'.    (25rt36x2)'  +  (ga^bx2)*  -  (i6asbxrf  = 
5ax(ab)*  +  3ax(ab)^  —  ^ax{db)^  =■  <\ax(ab)\    Ans. 

2.  Simplify  (azx2  —  a2x^. 

OPERATION. 

(a8x2  —  a2z3)*  =  (a2:c2/3  (a  —  x)*  =  ax  (a  —  x)^,    Ans. 

3.  Simplify  \/8i. 

OPEKATION. 


^/8i  =  ^27  x  3  =  ^27  ^3  ;        .-.     ^81  =  3  y3,     Ans. 
4.  Simplify  (aWx2  —  a5tfx)*. 

OPERATION. 

(aW-'-rt^x)'  ~  (azb3)'(aib-xi-a'-b\r)'  -  ab(aib°-x'i-a°bix)}>,    Ans. 


RADICALS.  127 

316.  To  Place  a  Rational  Factor  under  a  Radical  Sign  or 
Fractional  Exponent. 

Rule. — Involve  the  rational  factor  to  a  power  indicated  by 
the  reciprocal  of  the  fractional  exponent,  and  combine  it  with 
the  radical  factors,  if  there  be  any. 

5.  Eeduce  a2bc  to  a  radical  of  the  second  degree. 

OPERATION. 

a?bc  -  (tfbcf  =  [(tfbcf]*  =  (aW)",     Ans. 

In  this  case  there  were  no  radical  factors  with  which  to  combine  the 
rational  factors. 

6.  Eeduce  ab  (ax  —  z2)?  entirely  to  a  radical  form. 

OPERATION. 

ab  =  (a'262)l 
.-.    ab  (ax  —  a;2)*  =  [a262  [ax  —  x")]--  =  (aWx  —  a2&V)«,    Ans. 


317.  To  Reduce  Radicals  to  Like  Indices. 

Rule. — Eeduce  their  exponents  to  a  common  denominator. 

7.  Reduce  a*  xa§  x  eft  to  the  simplest  form. 

OPERATION. 

a*  =  a*,        ffl*  =  a",        a"  =  a*. 
.-.    a*a*a*  =  a1*'  =  a3. 

318.  To  Rationalize  one  Term  of  a  Fraction. 

Rule. — Multiply  both  numerator  and  denominator  by  that 
factor  which  will  render  the  required  term  rational. 


8.  Rationalize  the  denominator  of  vf. 

OPERATION. 

V*  =  Vt  -  V* x  2  -  3 V2>  Am- 


128 


RADICALS. 


9.  Kationalize  the  denominator  of 


V"  + 


Operation. — Multiply  both  terms  of  the  fraction  by  ^a—x,  which 
will  give 


a  —  x  a  —  x 


(a  +  g)*  +  {a  -  x)$ < 
(a  +  x)?  —  (a  —  x)* 


10.  nationalize  the  denominator  of 

OPERATION. 

Multiply  by  the  numerator,  and  we  have 


[(a  +  ,r)*  -t-  (a  —  .?■)']-  _  a  +  x  +  2  (a  +  x)*  (a  —  xfi  +  a .  —  x 
(a  +  x)  —  {a,  —  x)  2X 


a  +  (a-  —  x-)' 


,    Ans. 


EXAMPLES. 

Simplify  the  following: 


1.     a 


56  —  ]  a ■  —  3  (c  —  b)  +  2  [c  — 


a — 2b — c 


»(]'• 


2. 


1     a/«4  —  bA    4a     \ 

4  '      a2  _  tf     '  T  ' 

I    +  X 


f-511 


— S     ' 


I  —  X 


I   +  \/ 1  -f-  X         I  —  \A   +  # 

/a2.r  +  2rt.T2  +  a?U       /a2r  —  2ax~  +  a^\i 
\  a2  —  2Cix  -\-  x*  /         \  a2  +  2fu-  +  .t2  / 

5-     (5  +  2V3)  (5  —  2V3). 
V2  +  I2v6  —  4V10 


6. 


4Vs 


RADICALS. 


129 


4  —  V—  2 

—  2^/2 

(i  ±  v~3)3  +  (- 1  ±  V^y, 

a  a 

+ 


a  +  Va?  —  b2      a—  Va?  —  & 

(a  +  x)i  +  (a  —  z)*       (a  +  x)^  —  {a  —  .r)£ 
(a  +  &)*  —  (a  —  a)i       (a  +  a)*  +  (a  —  x)* 


—  sV^l  +    ioV—3    +  5^—3, 

IOV —  2 

—  sV^o 


5V—  2  2-\/  — 


2A/^ 

" X  — —   X 

3V  —  b  Va 


JZT. 

y        io 

5  x  \fy—  3-^3^—  3°- 


13-     4V  — 2  X  —3' 


2\/x 


14. 


15- 


2(1  —  y^)1 


2  (i    +    V^)- 


(i  +  ^)2-(i-^)2 
h  (I+ *)*  _  (1 -*)*}■--. r 


+ 


n(I+^-(i-#p-2r 


1  + 


16. 


T7- 


(i    +  .T2)* 


2  Vz  +  Vi  +  x2 

VaW  —  a2x3  +  V«5^4  —  a4^5  +  V(a  —  aO*. 


130 


1 8. 


19. 


RADICALS. 

x  +  a      x  —  a 
x  —  a      x  +  a 


x  —  a       x  +  a       x  +  a       x  —  a 
x  —  a      x  -f  a 

(cfi  —  x5  —  s^x  +  sax*  +  ioa3x2  —  iocfa?ft. 

(a3  —  $a2b  —  i2abx  +  6a2x  +  120a;2  +  yib2  +  S.r3 

—  12^  +  6«z— &»)*. 


ac 


22. 

23- 

24. 

25. 
26. 


V—  «c  x  —  V^c  X  —  ' 

a  {a  +  5)-1  +  fl  (a  —  &)-1 
a  (a  —  J)"1  —  6  (a  +  &)-*' 

121  °     1         1  S         1     ^  11 

Av  X  a^b^tr'  x  ami"c*  x  0"&V. 

W+w    m+n  m+n    m+n     m 

a  n  b  m  cmn  x  a  m  b  n  c". 


a  +  V—  ^      «  —  V— 
x 


X  —  V—  » be.        A, 


3,1 


1     71 


0 


Rationalize  the  denominators  of 

a  x2 

27-  — •  3°- 

31- 


29 


I 

Vb 

«  +  y- 
b3 

32. 


1 
X* 

+  r 

X2 

1 

—  y* 

1 

33- 


rr3  +  y* 


34-      —- 


$.  —  a*  x$  —  y? 

Note. — For  the  last  nine  examples  see  Art.  155,  and  examples  26-30, 


35-      — i- 

a~-  +  y~~* 


page  123. 


RADICALS.  131 


RADICAL      EQUATIONS. 

319."  Equations  containing  Radicals  may  often  be 
reduced  to  simple  equations  by  the  following 

Rule. — Involve  both  members  to  a  power  indicated  by  the 
reciprocal  of  the  radical  exponent. 

Notes.  —  i.  Before  involving  the  quantities,  it  is  generally  best  to 
clear  of  fractions,  and  transpose  the  terms,  so  that  one  member  of  the 
equation  shall  contain  but  one  radical  term. 

2.  In  reducing  such  equations,  the  student  should  remember  that 
any  of  the  methods  of  reduction  or  rationalization  of  radicals  may  be 
employed,  in  accordance  with  Axiom  i.     (Art.  38.) 


1.    Given  (12  +  xp  =  2  -+-  Vx,  to  find 


x. 


OPERATION. 

Squaring,  12  +  ;r  =  4  +  4\/%  +  x. 

Transposing  and  uniting  terms,  4yx  =  ^- 

•   Dividing  by  4,  and  squaring,  x  =  4,    Ans. 

2.  (2X  +  3)^04  =  7. 

3.  V'X  —  16  =  8  —  Vx. 

4.  V4«  +  x  =  2  (b  +.  a;)5  —  Vx. 

Vx  +  2ft  Vx  -f-  4ft 

Vx  +  b         Vx  +  3/y 

6        3^—i     __  t    [    V3X  —  1 
V3X  +  1  2 

-V  nix  —  V m  Vx  -f  m 

Vex  —  Vc         Vx  +  c 

yC         8.     (5  +  x)*  +  ^5^'  =  VZ. 

9.      V^  +-{t*  _  Vx-X\  -  3  (  _ lp  j  . 


CHAPTER    XI. 

EQUATIONS     OF     THE     SECOND     DEGREE. 

320.  Equations  of  the  Second  Degree  are  called 
Quadratics,  and  may  all  be  reduced  to  the  general  form 

Ax*  +  Bx  +  0  =  o,  (i) 

in  which  A  is  positive,  and  B  and  C  are  either  positive  or 
negative. 

321.  y!  cannot  be  o,  for  the  first  term  would  then  disap- 
pear, and  the  equation  be  of  the  first  degree. 

If  B  =  o,  the  second  term  disappears,  but  the  equation  is 
still  of  the  second  degree. 

If  C  =  o,  the  third  term  disappears,  and  the  equation, 
though  still  of  the  second  degree,  may  be  changed  to  one  of  the 
first  degree  by  simply  dividing  by  x. 

322.  Hence  we  have  but  two  cases  of  equations  of  the 
second  degree,  represented  by 

Ax2  +  C  =  o, 

which  is  called  the  incomplete  equation  of  the  second  degree, 
or  the  pure  quadratic;  and 

Ax*  +  Bx  +  0  =  o, 

called   the   complete  equation   of  the   second   degree  or   the 
affected  quadratic. 


THE     SECOND     DEGREE.  133 

323.  A  Complete  liquation  is  one  in  which  the 
series  of  powers  of  the  unknown  quantity  is  complete,  from  the 
highest  to  the  lowest;  as, 

axz  —  bx2  +  ex  +  d  =  o, 

in  which  the  exponents  of  the  powers  of  x  form  an  unbroken 
series,  from  3  to  o. 

324.  An  Incomplete  Equation  is  one  in  which  one 
or  more  terms  of  this  series  are  wauting ;  as, 

ax3  +  bx  +  c  =  o, 

in  which  the  term  containing  x2  is  wanting. 

325.  Dividing  equation  (i)  by  A,  we  have 

„       B  C  ,  , 

B  C 

Substituting  for  -j  and  -r,  2a  and  b, 

x*  +  tax  +  b  =  o,  (3) 

1 
a  form  to  which  every  quadratic  may  be  reduced,  in  which  a 

and  b  represent  any  quantities  whatever,  integral  or  fractional, 

positive  or  negative,  and  a  may  be  zero. 

326.  To  reduce  equation  (3),  if  b  =  a2,  we  may  take  the 
square  root  of  both  members  (Art.  156,  Cor.  1),  which  will  give 

x  +  a  =  ±0, 
and  x  =  —  a  ±  o. 

If  #  ^  «2,  we  may  always  add  to  b  such  a  quantity  as  will 
make  it  equal  to  a2.  This  quantity  will  be  a2 — b.  (Art.  1 1 2,  40.) 
But  to  preserve  the  equality,  the  same  quantity  must  be  added 
to  both  members  (Art.  38,  Ax.  1),  which  will  give 

x2  -f-  2ax  -\-  a2  =z  a2  —  b.  (4) 


134  EQUATIONS     OF 

327.  This  is  called   Completing  the  Square,   by 

which  is  meant 

Making  /he first  member  a  Perfect  Square. 

328.  It  will  be  observed  that  the  addition  of  a?  —  b  to 
both  members  is  the  same  as  transposing  b  and  adding  a2  to 
both  members.     Hence, 

To  make  the  first  member  of  a  quadratic  a  perfect  square, 
we  have  this 

Rule.  —  Transpose  the  absolute  term,  and  add  to  both 
members  the  square  of  half  the  coefficient  of  x. 

329.  Taking  the  square  root  of  equation  (4), 


x  4-  a  =  ±  V  a%  —  b, 


and  x  =  —  a  ±  V«2  —  0.  (A) 

If  a  =  o,  the  equation  is  incomplete,  and  (A)  becomes 


x=  ±  V—  0.  (B) 

As  equation  (A)  is  general,  applying  to  all  equations  of  the 
second  degree,  complete  and  incomplete,  aits  discussion  will 
furnish  all  the  principles  relating  to  the  roots  of  such  equa- 
tions.    These  may  be  enumerated  as  follows  : 

i°.    The  equation  has  two  roofs  (Art.  232), 


—  a  4-  Va~  —  b        and         —  a  —  \/a?  —  b. 

20.  The  sum  if  the  roofs  equals  —  2a.  or  the  coefficient  of  j 
first  power  with  its  sign  changed.     Hence, 

30.  The  sign  of  the  numerically  greater  root  will  be  unlike 
the  sign  of  the  second  term  or  2/u:     (Art.  106.) 

4Q.  The  product  of  the  roots  equals  b,  or  the  coefficient  of  x 
zero  power.     Hence, 

5°.  The  roots  will  hare  lil-e  siims  when  b  is  positive,  and 
■unlike  signs  when  b  is  negative.     (Art.  130.) 

6°.  The  rooh  will  be  equal  when  b  =  a2,  and  numerically 
equal  with  opposite  sigi<s  when  a  =  o. 


THE     SECOND     DEGREE.  135 

7°.  The  roots  will  he  real  when  b  is  not  greater  than  a2,  and 
imaginary  when  b  is  greater  than  a2. 

8°.  The  first  member  of  x2  +  zax  +  b  =  o  is  the  product 
of  (x  —  r)  (x  —  r'),  r  and  r'  representing  the  roots. 

Note. — The  student  will  easily  find  the  proof  of  these  propositions 
by  an  examination  of  the  roots  —  a  +  <\/a'2  —  b  and  —  a  —  y'a'2  —  b. 

330.  By  (8°)  we  have  a  ready  method  of  forming  an 
equation  having  any  given  roots. 

For  example,  to  form  an  equation  whose  roots  shall  be  3  and  —  5 
we  have  only  to  multiply  x  —  3  by  x  —  (—  5) ;  thus, 

(x  —  3)  (x  +  5)  =  x2  +  2X  -  15. 

This  put  equal  to  zero  is  the  equation  sought. 

Find  equations  having  the  following  roots: 

1.  7  and  —  2.  4.      —  a  and  —  b. 

2.  a  and  b.  5-     3  ±  V —  3. 

3.  a  and  —  b.  6.      —  2  ±  V—  r. 

331.  From  the  same  we  have  also  an  easy  method  of 
factoring  a  function  of  x  of  the  second  degree,  by  making  the 
function  =  o,  and  finding  the  values  of  x.  These  values 
connected  separately  with  x  by  the  sign  —  will  give  the  factors 
sough  t. 

332.  By  (50)  it  appears  that  both  roots  will  be  real,  or 
both  imaginary.  It  will  he  shown  hereafter  that  an  equation 
with  real  coefficients  cannot  have  an  odd  number  of  imaginary 
roots. 

Note. — Equation  (A)  should  be  regarded  as  a  formula  by  which  the 
roots  of  a  quadratic  can  be  written  without  writing  out  the  process  of 
completing  the  square.  The  student  should  commit  it  to  memory  for 
this  purpose. 

333.  The  formula  applies  equally  to  the  complete  and 
incomplete  equation,  reducing  in  the  latter  case  to  formula  (B). 


136  EQUATIONS     OF 

334.   When  the  reduction  of  the  equation  to  the  form 

X2  +  2dX  +  b  =  o 

involves   the  use  of  fractions,  this  may  be  avoided  by  the 
following  method. 

Let  the  equation  be  reduced  to  the  form 

ax2  -f-  bx  +  c  =  o. 
Multiplying  by  4a,  4a2x2  +  $abx  +  40c  =  o. 
Adding  b2  —  40c,  4a2x2  +  4Cibx  +  b2  =  b2  —  40c. 

Extracting  the  square  root, 


2ax  +  b  =  ±  Vb2  —  4«c, 
and  x  = ±  —  's/b2  —  4ac.  (C) 


Hence  we  have  for  completing  the  square  another 

Kule. — I.  Reduce  the  equation  to  the  form  ax2  +  bx-\-c  =  o. 

II.  Transpose  the  absolute  term,  and  multiply  by  4  times 
the  coefficient  of  x2. 

III.  Add  to  both  members  the  square  of  the  original 
coefficient  of  x. 

Note. — By  remembering  formula  (C),  it  may  be  used  instead  of  for- 
mula (A) ;  but  it  is  better  to  reduce  all  quadratics  to  the  form  (3), 
and  to  use  formula  (A). 

335.  After  reducing  the  following  equations  to  the  form 
(3),  let  the  student  answer  by  inspection  the  following  questions : 

1st.    What  is  the  sum  and  what  the  product  of  the  roots? 
2d.    Are  the  roots  equal  or  unequal  ? 
3d.     Are  they  real  or  imaginary  ? 
4th.  Have  they  like  or  unlike  signs? 
5  th.  If  like,  what  is  the  sign?     If  unlike,  what  is  the  sign 
of  the  greater  ? 

Note. — The  student  should  also  practice  the  substitution  of  the  roots 
obtained  to  verify  the  reduction.     (Art.  233.) 


THE     SECOND     DEGREE.  137 


Seduce  the  following: 

1.  7a;2  —  i6x  +  6S  =  (4X  —  2)2. 

2.  {x  +  sY  +  (z  -  5)2  =  6S- 

3.  (x  —  ay  +  (x  +  bf  —  o. 

x  +  2        4  —  :c         7 


4' 

5- 

6. 


K 


X  —  I  2X  3 

x  —  6       3  —  x  x 

x +  2        2  +  £         3 

3 2_  _  _5 

x  —  3       x  —  2        x  -\-  1 

xxi 

* 1 = 

x  +  2       2  —  2        ar  —  4 

1  1  1 

8.     + --  + =  o. 

x  —  a       x  —  0       x  —  c 

x  +  7.      x  —  3        17 
a;  _  3       a;  4.  3         4 

£  #  +  1        5 

10.     — —  +  ~—  =  -• 

x  -f  1  #  2 


II.       7 fw x  + 


(a;  —  £)  (a;  —  c)       (x  +  J)  (a;  +  c) 

12.  (z  —  1)  (x  —  2)  +  0  —  2)  (a  —  3)  =  (z  —  3)  (x  —  4) 

13.  x  [x  —  (a  +  b)~\  —  x  [a  —  (b  +  #)]  =  — 

1               2               3               11  -»_<. 

14. 1 1 ±—  = 

X  -\-  l         X  —  2         #  +  3  £+1 


< 


IS'      2         3 


*  -  -    - 


16.  a*  (1- I)  =  8  (3 +2). 

17.  ^T_^  +  ^-^==5-+aa;  +  ^-?. 

C  2  C-  C  2  C* 


tt2  4_   J2  —  2^T  _j_  x2  —. 


m-x 


2  ,.2 


11 


9.     w?a*2  +  ?;m  =  inwAjn  -+-  wa3. 


I 


138  EQUATIONS     OF 

PROBLEMS. 

r.  Divide  21  into  two  parts,  such  that  the  square  of  the 
less  shall  he  ^5-  of  the  square  of  the  greater. 

2.  Divide  1 4  into  two  parts,  such  that  their  product  shall 
be  -&  of  the  square  of  the  greater. 

3.  The  sum  of  the  squares  of  two  numbers  is  370,  and  the 
difference  of  their  squares  208.     What  are  the  numbers  ? 

4.  The  sum  of  the  squares  of  two  numbers  is  a%,  and  the 
difference  of  their  squares  is  b2.     What  are  the  numbers? 

5.  Find  a  number  such  that  when  divided  by  the  product 
of  its  digits,  the  quotient  will  be  2,  and  the  sum  of  its  digits 
is  9. 

6.  Divide  95  into  two  parts  whose  product  is  2146. 

7.  The  sum  of  two  numbers  is  100,  and  the- difference  of 
their  square  roots  is  2.     What  are  the  numbers  ? 

8.  A  man  travelled  60  miles,  and  if  he  had  travelled  1  mile 
an  hour  more,  he  would  have  required  3  hours  less  to  perform 
the  journey.     At  what  rate  did  he  travel  ? 

9.  A  boy  bought  oranges  for  30  cents.  If  he  had  bought 
5  more  for  the  same  sum,  they  would  have  cost  him  1  cent 
apiece  less.     How  many  did  he  buy  ? 

10.  Divide  a  into  two  parts,  such  that  the  sum  of  their 
square  roots  shall  be  s. 

11.  A  and  B  were  travelling  the  same  road  at  the  same 
rate.  At  a  certain  point  0,  A  overtook  C,  who  was  travel- 
ling at  the  rate  of  3  miles  in  2  hours,  and  two  hours  later  met 
D,  travelling  9  miles  in  4  hours.  B  overtook  C  5  miles  from 
0,  and  met  D  40  minutes  before  he  was  19  miles  from  0. 
Where  was  B  when  A  was  50  miles  from  0? 

12.  A  went  from  C  to  D,  travelling  a  miles  an  hour.  When 
he  was  b  miles  from  C,  B  started  from  D  towards  C,  and  went 

every  hour       of  the  distance  from  D  to  C.     When  B  had 

travelled  as  many  hours  as  he  went  miles  an  hour,  he  met  A. 
Find  the  distance  from  C  to  D. 


THE     SECOND     DEGREE.  139 

13.  From  1850  to  i860,  the  population  of  a  certain  town 
increased  1200,  which  was  a  percentage  of  gain  50  greater 
than  for  the  previous  decade.  From  i860  to  1870,  the  increase 
was  \  of  the  whole  gain  from  1840  to  i860,  and  the  increase 
from  1840  to  1870  was  bc/0.  What  was  the  population  in  1840, 
1850,  i860,  and  1870? 

14.  Divide  30  into  two  parts,  such  that  the  product  of 
their  squares  shall  be  46656. 

15.  A  body  of  men  were  formed  into  a  hollow  square 
3  deep,  when  it  was  observed  that  with  an  addition  of  52  to 
their  number  a  solid  square  might  be  formed,  of  which  the 
number  of  men  in  each  side  would  be  greater  by  2  than  the 
square  root  of  the  number  of  men  in  the  hollow  square.  What 
was  the  number  of  men  in  the  hollow  square  ? 

16.  A  looking-glass  12  by  18  inches  has  a  frame  of  uniform 
width,  and  of  the  same  area  as  the  glass.  What  is  the  width 
of  the  frame  ? 

17.  A  man  being  asked  his  own  age  and  that  of  his  son, 
answered :  "If  you  add  to  my  age  twice  the  square  root  of 
itself  and  subtract  24,  the  remainder  will  be  nothing.  The 
same  is  also  true  of  my  son's  age."    What  was  the  age  of  each  ? 

18.  The  product  of  two  numbers  is  m,  and  the  difference 
of  their  cubes  is  equal  to  n  times  the  cube  of  their  difference. 
What  are  the  numbers  ? 

19.  A  company  at  a  hotel  had  a  bill  of  $17.50  to  pay,  but 
before  it  was  paid  two  of  them  left,  when  those  who  remained 
had  each  to  pay  $1  more.  How  many  were  in  the  company  at 
first  ? 

20.  A  man  bought  a  certain  number  of  sheep  for  $300,  out 
of  which  he  reserved  15,  and  sold  the  remainder  for  $270, 
gaining  50  cents  a  head.  How  many  sheep  did  he  buy,  and 
at  what  price  ? 

21.  A  square  courtyard  has  a  rectangular  walk  around  it. 
The  side  of  the  court  is  2  yards  less  than  6  times  the  breadth 
of  the  walk ;  and  the  number  of  square  yards  in  the  walk 
exceeds  the  number  of  yards  in  the  perimeter  of  the  court  by 
92.    Find  the  area  of  the  court. 


140  EQUATIONS     OF 


HIGHER    EQUATIONS    SOLVED    BY    QUADRATICS 

336.  Many  equations  of  a  higher  degree  than  the  second 
may  be  reduced  as  quadratics.  These  may  all  be  reduced  to  the 
form 

xZn  +  2UXn  +  b  =  o, 

in  which  n  is  any  number,  integral  or  fractional. 

By  completing  the  square,  we  have  J 

x2n  +  2axn  +  a*  =  a2  —  b. 


Taking  the  root,  x*  +  a  =  ±  -\/a3  —  b. 


xn  —  —  a  ±  Va?  —  b 
and  x  =  (—  a  ±  Va~  —~b)K  (A) 

We  see  that  formula  (A)  may  be  used  for  writing  the 
values  of  xn,  from  which  the  roots  are  obtained  by  taking  the 
nth  root. 

Thus,     x  +  4-r*  —12  =  0,     in  which    n  =  £,    gives 

<r  =  —  2  ±  -y/i2  +  4  =  —  2  ±  4  =  2  or  -  6. 
x  —  4  or  36,     Ans. 

337.  Equationsof  this  sort  should  have  211  roots  (Art.  232), 
and  this  we  see  is  true,  for  xn  has  two  values,  and  x  is  the  ?cth 
root  of  these  values.  Therefore,  as  there  are  n  11th  roots  of  a 
quantity  (Art.  290),  x  has  in  values. 

338.  To  apply  this  to  the  last  equation,  it  will  be  observed 
that  it  is  of  the  second  degree  with  respect  to  z-,  and  z*  has 
two  values  ;  but  with  respect  to  x  it  is  of.  the  §  degree.  The 
degree  units  are,  so  to  speak,  only  half-units  ;  that  is,  the 
successive  exponents  differ  by  \  instead  of  1. 

So  also  the  roots  4  and  36  are  only  half-roots;  that  is? 
each  root  may  with  mathematical  accuracy  be  substituted 
in  two  wags,  while  only  one  of  these  will  satisfy  the  equation. 


THE     SECOND     DEGREE.  141 

Thus,  substituting  4  for  x,  we  have 

4  ±  42  =  12. 

We  get  the  double  sign  for  the  second  term,  because  the  square  root 
of  4  is  ±  2,  but  the  sign  +  only  will  satisfy  the  equation. 
If  we  substitute  36,  we  have 

36  ±  4-6  =  12, 

in  which  only  the  sign  —  will  give  a  true  result.     Hence  we  see  that  this 
equation  of  the  f  degree  has  two  half-roots. 

339.   Iii  the  form  x2n  +  2axn  +  b  =  o,  x  may  be  repre- 
sented by  a  binomial  or  a  polynomial;  as, 

(2X2  —  i)2  —  4  (2a;2  —  1)  —  21  =0. 

If  we  substitute  y  for  2X2  —  1 ,  we  get 

V1  +  41/  —  21  =  o, 
from  which  we  may  obtain  y  and  afterwards  x ;  but  it  is  better  to  consider 
2£2  —  1  the  unknown  quantity,  and  save  the  labor  of  making  the  sub- 
stitution.    By  so  doing,  we  get  directly  from  the  formula, 


2X1  —  1  =  2  ±  \/21  +4  =  2±5  =  7or 
x~  =  4  or  —  1. 
X  =  ±  2  or   ±  y/ —  I. 
Two  of  the  values  of  x  are  real  and  two  imaginary. 


EXAMPLES, 


I. 

x*  —  ar  =  a. 

2. 

x^  +  x™  =  b. 

3- 

Vet2  —  x2  +  2  (a2  - 

-  x2)  =  m  4-  n. 

4- 

1 

1                    1 

1  +  (a  —  of)*       V 

t   4-  (n  —  *\i          & 

S  4.    -1 --  I -I. 

1  +  (a  —  *)* 

5.  X3   +  X*    =    12. 

6.  as  4-  a*  =  6. 


+  1/        \    x    I  (n  —  i)' 


142 


EQUATIONS     OF 


X  — 


k-4 


x2  +  a  a 

IO-     (F+  ~dj^  ~  Ir1' 

ii.     (3a;  +  i)*  +  (3*  +  i)<  =  6. 


12.     Vi  +  x  (i  —  a/i  —  •'<')  =  Vi  —  a;  (i  +  a/ i  +  x). 

a:  —  \/a;  +  i        i 
i3- 7=  =  -• 

x  +  v  a;  +i        5 

14.  (tfi  +  xi)*  =  (a»  +  xyh 

15.  ir4  —  4a;3  +  5a'2  —  zx  =  o. 

16.  (1  -  a;  +  a*)*  -  (1  +  a  +  **)*  =  (1  -  3*)  (1  +  3*)- 

17.  x3  -}-  1  =  o. 

18.  £6  —  I  =  o. 


10 


x4  —  a' 


^  U'2  +  aa/  20 


21. 


x2  +  a2   '    \x*  + 
20.     1  +  (x  —  1)'  —  [1  +  {x  +  1)*]  =  1  —  V3- 

a;^  +  (x  —  a)'-'  _       n2a 
.'-•  —  (a;  —  a)¥       ^  —  a 

V«  +  x  +  \/«  —  .1;  / 

V«  +  a;  —  V«  —  a; 


23 


+  V] 


-vr 


THE     SECOND     DEGREE.  143 


THE     PROBLEM     OF     THE     LIGHTS. 

340.  Two  lights,  A  and  B,  of  given  intensities,  are  situated 
a  given  distance  apart.  Find  the  point  on  the  line  AB  where 
the  lights  will  give  equal  illumination. 

Let  U\  and  u-2  —  the  illumination  from  A  and  B  respectively,  at  any 
distance  x. 

Then,  since  by  the  principles  of  optics  the  illumination  varies 
inversely  as  the  square  of  the  distance  (Arts.  375-380), 

m  n 

U\   =  —, ;  and  «2  =   -=  , 

X2  X2 

in  which  m  and  n  are  constants,  depending  on  the  intensities  of  the  lights. 
If  we  make  x  =  1  in  each  of  these  equations,  we  have 

ii\  =  m        and        iii  =  n. 

Hence  we  see  that  m  and  n  represent  the  illumination  from  each  light 
at  a  unit's  distance. 

Making  x  =  the  distance  from  A  toward  B  to  the  point  of  equal 
illumination,  and  d  the  distance  between  the  lights,  we  have  for  that 

point, 

m  ,  "»■ 

ih  =  -5,        and        w2 


x*  (d  —  xf 

Since,  by  hypothesis,  U\  and  u.2  for  this  point  are  equal, 


m 


—  xV 


From  which  we  have 


x2       {d  -  x) 

d<\/m 

y/m  ±  y/n 

The  double  sign  in  the  denominator  gives  two  values  of  x,  and  there 
are  two  points  of  equal  illumination. 

341.   To  discuss  this  result,  assume  as  follows: 

1st.    Let    m  >  n. 

Then  both  values  of  x  will  be  positive,  one  less  and  one  greater  than 
d  ;  that  is,  one  point  will  be  between  A  and  B,  nearer  to  B,  and  the  other 
beyond  B.  This  is  evidently  as  it  should  be,  since  the  point  sought  must 
be  nearer  the  less  light. 

In  like  manner,  if  m  <  n,  the  point  between  A  and  B  comes  nearer 
to  A,  and  the  other  point  without  is  on  the  side  of  A. 


144  EQUATIONS     OF 

2d.     Let    m  =  n. 
This  gives  the  two  values 

x  —  Id        and        x  =  co  . 

The  first  of  these  is  evidently  correct,  since  if  the  lights  are  equal, 
the  point  of  equal  illumination  ought  to  be  equidistant  from  each. 

The  second  value  ( co  )  does  not  so  readily  appear  true  ;  but  when  we 
consider  that  ca  ±  d  =  co  (Art.  412),  we  find  this  point  also  equidistant 
from  the  lights. 

Another  question  might  arise  here,  whether  the  co  be  +  or  — ,  and 
if  +  why  not  also  — .  This  is  answered  by  observing  that  while  the 
lights  differ  in  intensity,  the  point  without  is  on  the  side  of  the  less 
light,  and  as  the  less  light  becomes  more  nearly  equal  to  the  greater, 
this  point  recedes  until,  when  the  difference  between  the  lights  is  infini- 
tesimal, its  distance  is  infinite,  but  still  on  the  side  of  the  light  which  is 
infinitesimally  less. 

3d.     Let    d  =  o,    and    in  ^  n. 

Then  both  values  of  x  are  o.  The  lights  are  then  at  the  same  point, 
and  no  place  except  this  point  will  be  equally  illuminated.     At  this 

point  (by  the  theory  that  gives  u  —  -  - 1,  if  X  =  o,  the  illumination  is 

infinite.  This  supposes  the  light  to  come  from  a  mathematical  point, 
which  has  no  dimensions,  a  thing  which  never  occurs,  since  the  source  of 
light  is  always  some  portion  of  matter  having  dimensions  Practically, 
therefore,  no  such  conditions  can  be  fulfilled;  but  it  furnishes  a  good 
illustration  of  the  general  character  of  mathematical  analysis,  which 
does  not  stop  with  the  possibilities  of  physical  conditions,  but  gives  the 
results  which  would  follow  from  given  laws,  if  the  physical  conditions 
could  be  realized. 

4th.    Let  n  be  negative. 

Then  the  values  of  x  are  imaginary,  showing  that  with  such  condi- 
tions there  is  no  point  of  equal  illumination. 

In  (3d),  if  the  conditions  could  be  fulfilled,  the  result  would  be  real, 
and  the  problem  admits  of  a  definite  answer.  In  (4th),  even  if  the 
conditions  could  be  fulfilled,  there  would  be  no  such  point  as  the  one 
sought,  and  the  result  is  therefore  imaginary.  To  fulfill  the  conditions 
of  n  being  negative,  it  would  require  that  B  should  diffuse  in  all  direc- 
tions some  light-absorbing  vapor,  by  which  the  light  from  A  should  be 
partially  neutralized,  the  law  of  its  diffusion  being  the  same  as  that  of 
the  diffusion  of  light.  But  even  thin  it  is  evident  there  would  be  no 
point  equally  illuminated  by  A  and  B.  If  m  and  n  are  both  negative,  the 
result  becomes  real. 


THE     SECOND     DEGREE.  145 

5th.     Let     d  =  o     and     m  =  n. 

Then  x  =  o  and  -•    This  last  value  is  indeterminate  (Art.  210),  and 
o 

represents  any  quantity  whatever,  which  is  evidently  a  correct  result, 

since  equal  lights  situated  at  the  same  point  should  illuminate  equally 

all  points  at  whatever  distance. 


SIMULTANEOUS  EQUATIONS  OF  THE  SECOND 
DEGREE. 

342.  The  general  equation  of  the  second  degree  between 
two  unknown  quantities  is 

ay2  +  bxy  +  ex2  +  dy  +  ex  +  /  =  o. 

This  equation,  combined  for  elimination  with  another  of 
like  form,  would  produce  an  equation  of  the  fourth  degree, 
but  combined  with  the  general  equation  of  the  first  degree, 

dy  -f  b'x  +  c'  =  o, 

gives  an  equation  of  the  second  degree.     Hence, 

343.  Simultaneous  equations  with  hvo  unknoivn  quan- 
tities, one  of  the  second  and  one  of  the  first  degree,  can  always 
be  reduced  as  quadratics. 

344.  There  are  also  certain  classes  of  simultaneous  equa- 
tions with  two  unknown  quantities  which  may  be  reduced  as 
quadratics,  when  both  are  of  the  second  degree.     These  are, 

1st.  Homogeneous  Equations. 
2d.    Symmetrical  Equations. 

345.  Hoinof/c neons  Equations  contain  the  same 
number  of  unknown  factors  in  each  term,  except  the  absolute 
term.  Such  equations  can  always  be  reduced  by  substituting 
for  one  of  the  unknown  quantities  the  product  of  the  other  by 
a  new  unknown  factor. 


146  EQUATIONS     OF 

346.  Symmetrical  Equations  have  the  unknown 
quantities  similarly  involved ;  as, 

x2  +  xy  +  y2  =  19. 

These  can  usually  be  reduced  by  substituting  for  x  and  y 
respectively  the  sum  and  difference  of  two  other  unknown 
quantities. 

347.  The  following  is  an  example  of  a  quadratic  and  a 
simple  equation: 


j  x2  +  y2  =  20 
(x  —  y   =     2 

(0 

(2) 

From  (2),                                           x  =  2  +  y 

(3) 

Substituting  in  (1),       (2  +  yf  +  y'2  =  20 

(4) 

Or,                                         y2  +  2y  =    8 

y  =  —  1  ±  \/s  +  1  =  —  r  ±  3 


y  =  2  or  —  4, ) 

Substituting  in  (2),  x  =  4  or  —  2.  \ 


Arts. 


348.  The  following  illustrates  the  reduction  of  homogeneous 
equations : 


-i 

X2  +  xy  =  10 

(1) 

wy  +  y1  =  x5 

(*) 

Put 

x  =  zy 

Then 

z-y*  +  zy-  =  10 

(3) 

Zy*  +  y*  =   15 

,(4) 

From  (3), 

y           Z-   +  2 

(5) 

From  (4), 

«2  =  

"           2  +   I 

(6) 

Equating, 

IO                 15 

22  +  2          2+1 

Hence, 

3  =  %   or  — 

1 

Substituting 

|  for  2  in  (5)  or  (6),  we  have, 

y  -  ±  3 

rr  =  ±  2 

Substituting 

—  1  for  2  in  (5)  or  (6) 

y  =  ±  00 

a;  =  =?  <» 

we  hav 

e 

THE     SECOND     DEGREE.  147 

Since  the  combination  of  these  equations  of  the  second  degree  give 
an  equation  of  the  fourth  degree,  there  should  be  four  roots.  Two  of 
them  in  this  case  are  oo  .  Substituting  +  oo  for  x,  and  —  oo  for  y,  we 
have, 

CO2    —    OO2   =    IO 

OO2    —    OO-    =    15 

results  which  are  consistent,  as  will  be  seen,  Art.  412. 

349.  As  an  example  of  Symmetrical  Equations,  we  have, 

(  z8  +  V1  =  13  (1) 

3'    1  xy  =    6  (2) 

Multiplying  (2)  by  2,  and  adding  it  to  and  subtracting  it  from  (1), 

X2  +  2xy  +  y2  =  25 

x2  —  2xy  +  if  =    1 

x  +  y  =  ±  5 

x  -  y  =  ±  1 

Adding,  2x  =  ±  6 

Subtracting,  2y  =  ±  4 

x  =  ±  3  )    , 

\  Ans. 
y  —  ±  2  S 

350.  Symmetrical  Equations  with  two  unknown  quantities 
may  frequently  be  reduced,  when  one  is  of  the  first  degree 
and  the  other  of  a  degree  higher  than  the  second. 


3 


4-     ■ 

;  X   +  y   =       8 

(I) 

.  x9  +  yz  =  152 

(») 

Let 

X  =  « +  V 
y  —  u  —  v 

Then 

x  +  y  =  211  —  8 

And 

u  =  A 

Substituting  in  (2),      (4  +  vf  +  (4  —  -y)3  =  152 
Developing  and  reducing,  241;2  =    24 

v  =  ±  1 
a;  =  M  +  «  =  4±i  =  5  or  3 
y  =  u  —  !)  =  4ti  =  3  or  5 


148  EQUATIONS     OF 

Ileduce  the  following  by  these  or  other  methods,  as  shall 
be  found  feasible : 

5.  x  —  y   —      4,  17.     x3y2  +  x2y3  =  —  4, 
Xs  —  y3  =  124.  x2y  +  xy2   =  2. 

6.  x  +  y   =    9,  18.     Vz  +  V#  =    6, 
z2  +  */2  =  53-  x  +      y  =  20. 

7.  x  +  */  =  a,  i9.     Vx  +  Vy  =    4, 
&  +  y*  =  h.  x%  +  yl  =  28. 

8.  a:   +  ?/    =5,  20.     £B  —  #   =     1,  

^  +  f  =  97-  a4  —  y*  =  15. 

9.  232  +■  xy  =14,  21.     x2  +  3.r#  —  y2  =  36, 

2?/2  —  ##   =    12.  3^  +   21J    =    I  6. 


10. 

II. 

x3y  —    y  =  21, 
a;2?/  —  .ry  =    6. 

z2+2Z#  +  iy  +  3.T=73, 

22. 

*4_y 

y+  x 

x  +  xy  +  y  =  14- 

2/2  +  3!/  +  «        =44- 

23- 

s2  +  xy      =  40, 

•■         12. 

13- 

#2y2  _|_  ^^   =    12, 
2-  +  y       =     I. 

x  —y  —    2, 

x3  —  y3  =  56. 

24. 
25- 

3»y—  2«/2  =  27. 
x2  +  2.r?/  4-  y2  =  81, 
a;2  —  22/y  +  ?/2  =     9. 
x2  —  gy2  =  16, 
#3  —    ?/2  =  24. 

14. 

a-2  -  if  =  8, 

26. 

a;2  (x  —  y)       =    4, 

T 

27. 

x2(2x  +  $y)  =  28. 
xy2  +  2$  =  24, 

v^2  +  y2  =     ~' 

15. 

x  +  y       x  —  y       10 
x  —  y       x  +  y —   3 ' 

28. 

xj/2  +  x    =40. 
x2  +  y2          =13, 

a2  +  y2  =  5. 

zxy  —  x  —  y  —     7. 

16. 

1        1         1 
x      y~  a' 

29. 

x  +  y  __  7 

x  —  y  '   3' 

x2  +  y2  =  Z>. 

x—if=  1. 

3°- 

a;  +  y         2X         a?  — 

x2y 

3. 

V          z  +  y      y3- 

x2y  ~ 

5' 

y2- 

-x2  = 

5- 

THE     SECOND     DEGREE.  149 

PROBLEMS. 

i.  Divide  a  into  two  parts,  whose  product  shall  equal  b. 

Let  x  and  y  represent  the  parts.     Then,  by  the  conditions, 
x  +  y  -  a  (i) 

xy  =  b  (2) 

From  (1),  y  —  a  —  x 

Substituting  in  (2),  (a  —  x)  x  =  b 

x*  —  ax  +  b  =  o 


x  =  \a±  y 0 

/a? 
The  two  parts  are  ^a  +  JU b 


A2       . 
and  ia  —  y b, 


a? 
from  which  it  appears  that  if  b>  — ,  the  values  are  imaginary.     Hence, 

Cor. —  The  product  of  two  quantities  cannot  be  greater  than 
the  square  of  half  their  sum. 

Or,  The  jjroduct  of  the  tioo  parts  of  a  given  quantity  is 
greatest  when  those  parts  are  equal. 

2.  Find  three  numbers,  the  difference  of  whose  differences 
is  8,  their  sum  is  41,  and  the  sum  of  their  squares  699. 

3.  Find  three  numbers,  the  difference  of  whose  differences 
is  5,  their  sum  is  44,  and  their  continued  product  is  1950. 

4.  The  fore-wheel  of  a  carriage  makes  6  revolutions  more 
than  the  hind-wheel  in  going  120  yards,  but  if  the  circumfer- 
ence of  each  wheel  be  increased  one  yard,  it  will  make 
4  revolutions  more  than  the  hind-wheel  in  the  same  distance. 
"What  is  the  circumference  of  each  wheel  ? 

5.  What  number  being  divided  by  the  product  of  its  two 
digits  gives  the  quotient  2,  and  if  27  be  added  to  the  number, 
the  digits  will  be  inverted  ? 


150         EQUATIONS    OF    THE    SECOND    DEGREE. 

6.  The  difference  between  the  hypothenuse  and  base  of  a 
right-angled  triangle  is  6,  and  the  difference  between  the 
hypothenuse  and  perpendicular  is  3.     What  are  the  sides  ? 

7.  A  and  B  put  out  at  interest  different  sums  amounting 
to  $200.  B's  rate  of  interest  was  \c/0  more  than  A's.  At  the 
end  of  5  years,  B's  accumulated  simple  interest  was  $4  less 
than  the  double  of  A's.  At  the  end  of  10  years,  A's  principal 
and  interest  was  f  of  B's.  What  was  each  sum  and  rate  per 
cent? 

8.  Two  partners,  A  and  B,  gained  $140  in  trade.  A's 
money  was  3  months  in  trade,  and  his  gain  was  £60  less  than 
his  stock,  and  B's  money,  which  was  $50  more  than  A's,  was 
in  trade  5  months.     What  was  each  man's  stock? 

9.  Find  two  numbers,  the  difference  of  whose  squares  is 
m%,  and  which  being  multiplied  respectively  by  a  and  b,  the 
difference  of  the  products  is  n2. 

10.  Divide  a  and  b  each  into  two  parts,  such  that  the 
product  of  one  part  of  a  by  one  part  of  b  shall  be  m,  and  the 
product  of  the  remaining  parts  n. 

11.  What  is  the  side  of  a  cube  which  contains  as  many 
units  of  volume  as  there  are  linear  units  in  its  diagonal  ? 

12.  Find  two  numbers  whose  sum,  product,  and  sum  of 
their  squares  shall  be  equal  to  each  other. 

13.  Find  two  numbers  whose  sum,  product,  and  difference 
of  their  squares  are  equal  to  each  other. 

14.  Find  two  numbers  whose  product  equals  the  difference 
of  their  squares,  and  the  sum  of  their  squares  equals  the 
difference  of  their  cubes. 

15.  The  product  of  the  sum  and  difference  of  two  numbers 
is  8,  and  the  product  of  the  sum  of  their  squares  and  the 
difference  of  their  squares  is  80.     What  are  the  numbers  ? 

16.  A  and  B  bought  600  acres  of  laud  for  $600,  each  pay- 
ing $300.  In  dividing,  A  took  the  best  land  and  paid 
75  cents  an  acre  more  than  B.  HowT  much  land  did  each  get 
and  at  Avhat  price  ? 


CHAPTER    XII. 

INEOUATIONS. 

351.  An  Inequation  is  an  expression  of  inequality 
between  two  quantities ;  as,  a  >  b  (read,  "  a  is  greater  than  #") ; 
x  <  y  (read,  "  x  is  less  than  y  "). 

The  quantity  on  the  left  of  the  sign  is  called  the  First 
Member,  and  the  one  on  the  right  the  Second  Member 

of  the  inequation. 

352.  Algebraically,  a  negative  quantity  is  said  to  be  less 
than  zero  (Art.  104) ;  and  of  two  negative  quantities,  that 
which  is  numerically  greater  is  algebraically  less.     Therefore, 

If  a  —  b  >  o,  a  >  b, 

and  if  a  —  b  <  o,  a  <  b. 

353.  In  the  transformation  of  inequations,  it  is  necessary 
to  observe  when  the  sign  of  inequality  will  be  reversed. 

When  this  sign  is  reversed,  the  tendency  of  the  inequation 
is  said  to  be  changed. 

Thus,  a > b  and  c> d,  are  inequations  of  the  same  tendency,  and 
and  a  >  b  and  c  <  d  are  of  opposite  tendency. 

354.  The  tendency  of  an  inequation  is  not  changed, 

1.  By  any  like  operation  upon  both  members,  except 
changing  their  signs. 

2.  By  adding  or  multiplying  by  the  corresponding  members 
of  an  inequation  of  the  same  tendency:  Provided  that  in 
multiplying,  the  signs  of  the  members  be  not  changed. 


152  INEQUATIONS. 

3.  By  subtracting  or  dividing  by  the  corresponding  members 
of  an  inequation  of  opposite  tendency :  Provided  that  in 
dividing,  the  signs  of  the  members  be  not  changed. 

355.  The  tendency  of  an  inequation  is  changed, 
By  changing  the  signs  of  both  members.     (Art.  104.) 
For,  2  <  3,  but  —  2  >  —  3. 

356.  The  tendency  of  an  inequation  becomes  doubtful, 

1.  By  subtracting  or  dividing  by  the  corresponding  mem- 
bers of  an  inequation  of  the  same  tendency. 

2.  By  adding  or  multiplying  by  the  corresponding  members 
of  an  inequation  of  opposite  tendency. 


For  it  is  evident  that  if  a  <  b  and  e 

<d, 

a 

-0§l 

-d, 

and 

a  <  b 

c  >  d 

Thus,             (1.)        5  <  15 

and 

3<6. 

Subtracting, 

2  <  9. 

Dividing, 

5  <---  5 
3  ^- 1- 

(2.)         5  <  15 

and 

2  <  12 

Subtracting, 

3  =  3- 

(3)         5  <  15 

and 

i<3- 

Dividing, 

5  =  5- 

(4.)         5  <  6 

and 

K4. 

Dividing, 

5>l- 

Subtracting, 

4>2. 

357.  The  Red}(ction  of  an  Inequation  consists 
in  so  transforming  it  that  the  unknown  quantity  may  stand 
alone  as  one  member,  while  the  other  member  contains  only 
known  quantities,  the  value  of  which  is  a  limit  to  the  value  of 
the  unknown  quantity  in  one  direction.  If  two  inequations 
containing  the  same  unknown  quantity  can  be  reduced  with 
opposite  tendencies,  limits  in  both  directions  will  be  found. 


INEQUATIONS.  153 


EXAMPLES. 
X  X 

i.  Given  — (-  3  >  ^  4-  2,  to  find  a  limit  for  x. 
4  o 

Solution. — Clearing  of  fractions,        6a;  +  72  >  421  +  48. 
Transposing,  2X  >  —  24. 

Dividing  by  2,  a;  >  —  12,    Ans. 

2.  Given  ■}  _  ]■  to  find  limits  for  z. 


2£  +   4   >    16  —    2X 

Solution. — Transposing  and  uniting  terms, 

2a;  <  10  and  4.?'  >  12. 

x  <    5  and  x  >    3,     ^4ws. 

3.  Given  32;  +  72;  —  30  >  10,  to  find  a  limit  for  x. 

4.  Given  x  4-  |#  —  |x  >  4,  to  find  a  limit  for  x. 
ax  ,  _    a2 


5.  Given 

6.  Given 

7.  Given 


^2 
4-  bx  —  a#  >  — 
5  5 

bx  b2 

—  —  ax  4-  ab  <  — 

I  7  7 


to  find  limits  for  x. 


$x  —  4<    x  +    6)   to  find  an  integral  value 
53  +  7  >  3X  +  l3)  of  &• 

£(a;  +  2)  +  !£  <  i  (^—4)+3  )  to  find  an  inte- 
^  (a;  4-  2 )  4-  J 2;  >  -2-  (z  4- 1 )  +  i  j    gral  val  ue  of  x. 


8.  A  certain  integral  number,  doubled  and  diminisbed  by 
7,  is  greater  tban  29  ;  and  3  times  the  number  diminished  by 
5  is  less  than  double  the  number  increased  by  16.  What  is 
the  number  ? 

9.  A  boy  sold  a  number  of  apples,  such  that  triple  the 
number  increased  by  2  exceeds  double  tbe  number  increased 
by  61  ;  and  5  times  the  number  diminished  by  70  is  less  than 
4  times  the  number  diminished  by  9.    How  many  did  he  sell  ? 

10.  The  sum  of  two  whole  numbers  is  25.  If  the  greater 
be  divided  by  the  less,  the  quotient  will  be  greater  than  § ; 
and  if  the  less  be  divided  by  the  greater,  the  quotient  will  be 
greater  than  |.     What  are  the  numbers  ? 


CHAPTER    XIII. 

RATIO     AND     PROPORTION. 

358.  A  Ratio  is  the  quotient  arising  from  the  division 
of  one  quantity  by  another. 

The  sign  of  division  commonly  used  to  express  ratio  is  the 
colon  (:),  as  a  :  b,  which  means  the  ratio  of  a  to  b.    But  a  -=-  h 

and  7  express  the  same  thing. 

359.  A  Proportion  is  an   equality  of  ratios,  or  an 

a        c 
equation  each  of  whose  members  is  a  ratio ;   as  ■=■  =  -,   or 

b        a 

a  :  b  =  c  :  d. 

Notes. — i.  The  double  colon  (:  :)  is  frequently  used  as  the  sign  of 
equality  in  a  proportion,  but  without  good  reason. 

2.  It  is  also  questionable  whether  it  were  not  better  to  express  ratio 
in  ike  form  of  a  fraction,  instead  of  using  a  special  sign  of  division. 

360.  The  First  Term  of  a  ratio  is  called  the  Ante- 
cedent; the  Second  Term  the  Consequent. 

361.  The  First  and.  Last  Terms  of  a  proportion  are 
called  the  Extremes ;  the  Second  and  Third  the 
Means. 

362.  When  the  same  quantity  is  used  for  both  ///ran*,  it  is 
called  a  Mean  Proportional  between  the  other  two:  and 
the  last  term  a  Third  Proportional  to  the  other  two. 
In  this  case  the  three  quantities  are  said  to  be  in  Continued 
Proportion  ;  as,  a  :  b  =  b  :  c. 

363.  A  Compound  Ratio  is  the  product  of  two  or 

a       c       in 
more  ratios ;  as,  T  x  -,  x  —  • 
7        o       a       n 


RATIO     AND     PROPORTION.  155 

364.  A  Compound  Proportion  is  one  in  which 
there  is  a  compound  ratio. 

Theorem  I. 

365.  In  any  proportion,  the  product  of  the  extremes  equals 
the  product  of  the  means. 

Demonstration. — Let  a  :b  =  c  :  d, 

a       c 

0r  r.  =  *■ 

b       d 

Clearing  of  fractions,     ad  =  be. 

Cor.  i. — If  three  terms  are  in  continued  proportion,  the 
square  of  the  mean  is  equal  to  the  product  of  the  extremes. 

For,  if  a  :  b  =  b  :  c,        .'.     b2  =  ac. 

Cor.  2. — A  mean  proportional  between  tivo  quantities  is 
the  square  root  of  their  product. 

For,  if  b2  =  ac,    .'.    b  —  ^/ac. 

Cor.  3. — Tlie  product  of  the  extremes  divided  by  one  mean 
equals  the  other  mean,  and  the  product  of  the  means  divided  by 
one  extreme  equals  the  other  extreme. 

For,  if    ad  =  be,        .'.     a  —  — ,         and        d  =  — . 
d  a 

ad  ad 

Also  0  =  — ,         and        c  =  — . 

c  b 


Theorem  II. 

366.  If  the  product  of  two  quantities  be  equal  to  the  pro- 
duct of  two  other  quantities,  the  factors  of  either  product  may 
be  made  the  means  with  the  factors  of  the  other  product  for 
the  extremes  of  a  proportion. 

Dem. — Let  ad  =  be ; 

Dividing  by  bd,  -  =  -  ; 

Or,  a :  &  =  c  :  (?. 


156  RATIO     AND     PROPORTION". 

Cor.  i. — If  the  factors  of  one  product  are  the  same,  that 
factor  is  a  mean  proportional  between  the  other  tico. 

Cor.  2. — Tli e  means  or  extremes  may  change  places  ;  for 
the  order  in  which  the  factors  arc  t alien  is  not  material. 

If      ■  a  :b  =  c  :  d, 

Then  a  :  c  =  b  :  d. 

Tliis  change  is  called  Alternation, 


Theorem  III. 

367.   A  proportion  will  remain  true  if  loth  of  its  ratios  be 
inverted. 

Dem. — If  two  quantities  are  equal,  their  reciprocals  will  be  equal. 

That  is,  if  %  =  e-, 

o        a 

Then  *   =  * 

a        c 


This  is  called  proportion  by  Inversion, 


Theorem  IV. 


368.  The  sum  or  difference  of  the  terms  of  the  first  ratio 
is  to  either  the  antecedent  or  consequent  of  that  ratio,  as  the 
sum  or  difference  of  the  terms  of  the  second  ratio  is  to  the 
antecedent  or  consequent  of  the  second  ratio. 


Dem.- 

-Let 

a  :  b  =  c  :  d, 

Or, 

a      c 
b~d' 

Then 

a             c 
b±1  =  d±X' 

Or, 

a  ±b       c ± d 
~b~       ~d~' 

Again, 

b  _d 

a  ~~  c  ' 

b             d 

I  ±  -  =  i  ±  - 
a            c 

RATIO     AND     PROPORTION.  157 


a  ±  b       c  ±  d 

a             c 

Hence, 

a ±b :b = c ± d 

d; 

And 

a ±b : a  —  c ± d 

c. 

This  is  called  proportion  by  Composition  or  Divi- 
sion ;  the  former  when  the  sum,  the  latter  when  the 
difference  is  used. 

Cor.  i. — If  four  quantities  are  in  proportion,  the  sum  of 
the  first  and  second  is  to  the  sum  of  the  third  and  fourth  as 
the  difference  of  the  first  and  second  is  to  the  difference  of  the 
third  and  fourth. 

For  by  Alternation,  a  +  b  :  c  +  d  =  a  :  c, 

And  a  —  b  :  c  —  d  —  a  :  c. 

But  a  :  c  =  b  :  d; 

Therefore,  a  +  b:  c  +  d  =  a  —  b  :  c  —  d; 

Also  a  +  b  :  a  —  b  =  c  +  d  :  c  —  d;  hence 

Cor.  2. — The  sum  of  the  first  and  second  is  to  their  differ- 
ence as  the  siim  of  the  third  and  fourth  is  to  their  difference. 

Cor.  3. — In  any  number  of  equal  ratios,  the  sum  of  the 
antecedents  is  to  the  sum  of  the  consequents  as  any  one 
antecedent  is  to  its  consequent. 

For  if  -  =  -  =  i  =  r; 

b      d      f     r' 

a  +  c  +  e  +  &c.  a       c 

■  •     iT7-^ * i —  =  r  =  r  =  &c.    (Art.  200.) 

b  +  d  +  J  +  &c.  b  v  ' 

369.   These  changes  may  all  be  expressed  as  follows  : 

If  four  quantities  are  in  proportion,  they  will  be  in 
proportion  by  Alternation,  Inversion,  Composi- 
tion, or  Division. 


158  RATIO     AND     PBOPDRTION, 


Theorem  V. 

370.  If  four  quantities  are  in  proportion,  the  proportion 
will  he  true  if  both  antecedents,  both  consequents,  both  terms 
of  either  ratio,  or  ail  the  terms  be  multiplied  by  the  same 
quantity. 

Let  the  student  prove  each  of  these  by  the  properties  of 

ratios. 

Theorem  VI. 

371.  If  both  antecedents  or  both  consequents  be  increased 
or  diminished  by  adding  or  subtracting  quantities  having  the 
same  ratio  as  the  antecedents  or  consequents,  the  results  will  be 
in  proportion  with  the  antecedents  or  consequents,  or  with 
each  other. 


Dem. 

—Let 

a  :  b  =  c  :  d  ; 

Then 

a               c 

-  ± m   = -±m; 

b                d 

Or 

a  ±mb       c  ±  md 

b                d 

Also 

b              d 

-  ±  n  =  -  ±n; 
a             c 

Or 

b  ±  na       d±  nc 

II                     c 

•"• 

a  ±  a ib 

:  e  ±  md  =  l  ~  na 

Theorem  VII. 

d  ±nc. 


372.  A  proportion  is  not  destroyed  by  affecting  each  term 
by  the  same  integral  or  fractional  exponent. 

Let  the  student  furnish  the  proof. 

Note. — The  ratio  of  the  squares  of  two  quantities  is  called  the 
Duplicate  ll<(tio,  and  the  ratio  of  the  cubes  the  Triplicate 
llatio. 

Also  the  ratio  of  the  square  roots \b  called  the  Sub-duplicate,  and 
of  cube  roots  the  Sub-triplicate  ratio. 


RATIO     AND     PROPOETION.  159 


Theorem  VIII. 

373.    Tlie  products  of  the  corresponding  terms  of  any  num- 
ber of  proportions  are  proportional. 

This  is  only  the  multiplication  of  several  equations,   member   by 
member, 


As, 


a  _  c 

b  ~  d' 
m  _  x  _ 

n  _  y' 
am  _  ex 
bn  ~  dy 

374.  When  two  quantities  have  the  same  ratio  as  the 
reciprocals  of  two  other  quantities,  they  are  said  to  be 
Reciprocally  Proportional ;  as, 

a  :  0  =  -  :  -=• 

c    d 

This  may  be  written  a  :  b  —  d  :  e,  in  which  the  terms  of  the  second 
ratio  are  inverted  ;  hence  they  are  also  said  to  be  inversely  proportional. 

This  has  no  meaning  unless  a,  b,  c,  and  d  are  so  related  that  c  in 
some  way  belongs  to  a,  and  d  to  b.  Otherwise  the  order  d :  c  would  be 
no  more  an  inverted  order  for  the  second  ratio  than  c  :  d. 

ILLUSTRATION. 

Suppose  two  men  travel  at  different  rates  the  same  dis- 
tance.    Then  the  times  of  travelling  would  he  different. 

Let  r  and  r'  be  the  rates,  t  and  f  the  corresponding  times.  We 
shall  then  have 

r:r'  =  l^  =  f:t. 

The  terms  of  the  second  ratio  must  here  be  taken  in  inverse  order, 
or  the  reciprocals  must  be  used.  If  they  travel  at  the  same  rate  but 
different  times  and  distances,  we  have,  putting  d  and  d'  for  distances, 

t  :t'  -  d:  d'. 

In  this  case  tlie  quantities  are  said  to  be  Directly 
Proportion &l ;  in  the  former  they  are  Reciprocally 
or  Inversely  Proportional, 


ICO  RATIO     AXD     PROPORTION". 

PROBLEMS. 

i.  Find  a  third  proportional  to  25  and  50. 

2.  The  last  three  terms  of  a  proportion  are  36,  24,  and  16. 
What  is  the  first  term  ? 

3.  Find  a  mean  proportional  between  5  and  45  ? 

4.  If  a  men  working  b  hours  a  day  can  complete  a  certain 
work  in  c  days,  in  how  many  days  can  a  men  working  V 
hours  a  day  do  it  ? 

5.  Two  wagons  with  their  loads  have  their  weights  in  the 
ratio  of  4  to  5.  Parts  of  their  loads  in  the  proportion  of  6  to 
7  being  removed,  they  weigh  in  the  ratio  of  2  to  3,  and  the 
sum  of  their  weights  is  then  10  tons.  What  were  the  weights 
at  first  ? 

6.  A  starts  to  travel  from  Cto  D,  and  3  hours  afterwards 
B  starts  from  D  towards  C,  travelling  2  miles  an  hour  more 
than  A.  When  they  meet  the  distances  they  have  travelled 
are  in  the  ratio  of  13  to  15.  If  A  had  travelled  5  hours  less 
and  B  had  gone  2  miles  an  hour  faster,  they  would  have  been 
in  the  ratio  of  2  to  3.  How  many  miles  did  each  go,  and  how 
long  did  each  travel  before  they  met  ? 

7.  Find  3  numbers  such  that  if  6  be  added  to  the  first  and 
second  the  sums  will  be  in  the  ratio  2  to  3,  and  if  5  be  added 
to  the  first  and  third  the  sums  will  be  in  the  ratio  of  7  to  11, 
but  if  36  be  subtracted  from  the  second  and  third,  the  remain- 
ders will  be  as  6  to  7. 

8.  Find  the  two  numbers  whose  sum  is  to  the  less  as  5  to 
2.  and  whose  difference  multiplied  by  the  difference  of  their 
squares  is  135. 

9.  A  certain  number  has  3  digits.  The  first  is  to  the 
third  as  16  to  6,  the  third  to  the  second  as  1  to  2,  and  the  sum 
of  the  digits  is  17.     What  is  the  number? 

10.  Find  three  numbers  whose  sum  is  to  the  first  as  6  to  1, 
to  the  second  as  3  to  1,  and  to  the  third  as  2  to  1. 

11.  Find  two  numbers  the  ratio  of  whose  difference  to  their 
sum  is  m,  and  the  ratio  of  the  difference  of  their  squares  to 
the  sum  of  their  squares  is  n. 


VARIATION.  161 


VARIATION. 

375.  The  'Relations  of  Quantities  are  sometimes 

expressed  by  saying  that  one  varies  directly  or  inversely 

as  the  other,  or  as  the  square  or  cube,  or  some  other  function 
of  the  other. 

376.  The  Sign  ,of  Variation  (  oc  )  is  used  to  express 
these  relations  ;  as,  x  oc  y,  which  is  read  "  x  varies  as  y,"  and 
means  that  x  and  y  are  such  functions  of  each  other  that 
their  ratio  remains  the  same,  and  that  whatever  changes  may 
take  place  in  the  quantities  themselves,  the  increase  or 
decrease  of  one  must  always  be  proportional  to  the  increase 
or  decrease  of  the  other. 

377.  The  same  thing  may  be  expressed  in  other  ways. 

Let  m  represent  a  constant  quantity  ;  that  is,  a  quantity  which  in 
the  same  discussion  does  not  change  its  value,  and  we  may  write 

x  oc  y  ;    x  =  my  ;     or     -  =  m, 
each  of  which  expresses  the  same  relation. 

378.  If  xt  and  x2  represent  two  different  values  of  x,  and 
yx  and  y^  the  corresponding  values  of  y,  then  the  same  thing 
is  also  expressed  by  the  proportion 


xx 

x2  = 

=  l/i 

y*> 

OT 

Xi   _ 

x$ 

-  I1. 

y* 

379. 

If 

X 

oc 

i 

y 

then 

x  - 

=  m 

i 

x  - 

y 

= 

m 

~y 

or 

* 

x 

i 
^y 

= 

m, 

or  xy  =  m. 

In    this    case  x  varies   reciprocally  as  y,   while  in 
x  oc  y,  x  is  said  to  vary  directly  as  y. 


162  VARIATION. 

380.    From  x  oc  -   we  have  the  proportion 

ii 

Xi  '.  x%  =       :       5 

2/1  y* 

in  which  x  and  y  are  said  to  be  reciprocally  proportional,  or 

xx  :x.2  =  y2:  ?/, ; 
in  which  they  are  inversely  proportional.     (Art.  374.) 

EXERCISES. 

381.   Form    the    equations    and    proportions    which    are 
implied  in  the  following : 


1.  x  oc  y2. 

x  =  my3 ;        —  =  m  ;        xx  :  xa  =  yx9  :  y.,2,    Ans. 

2.  x  oc  — 2-  4.     x  oc  y2  +  ^3. 

T 

3.  x  oc  tf  +  y.  5.     x  oc 


2/  +  r 


6.  If  a;  oc  y,  and  a;  =  3  when  y  =  5,  what  is  the  value  of 
y  when  k  =  15  '?  When  a;  =  7  ?  What  of  x  when  3/  =  20  ? 
When  y  =  8  ? 

7.  If  a;  oc  ?/2,  and  a;  =  1  when  y  =  5,  what  is  the  value  of 
a;  when  _?/  =  7  ?     3?     2?     o? 

8.  If  z  oc  mi-  +  y,  and  2  =  3  when  a;  =  1  and  y  =  2, 
and  z  =  5  when  a;  =  2  and  y  =  3?  what  is  the  value  of  m  ? 

9.  If  x2  oc  y3,  and  a;  =  2  when  y  =  3,  what  is  the  value  of 
y  in  terms  of  x  ? 

10.  If  y  oc  v  4-  w,  r  oc  x,  and  w  oc  -;  and,  if  y  =  4  when 

a:  =  1,  and  7/  =  5  when  x  =  2,  what  is  the  value  of  y  in  terms 
of  a;. 

n.  If  a  body  falls  192  inches  the  first  second  and  the  dis- 
tance it  falls  varies  as  the  square  of  the  time,  how  far  will  it 
fall  in  10  seconds  ?     In  how  many  seconds  will  it  fall  400  ft.  ? 


C  H  APTER    XIV. 

PERMUTATIONS     AND     COMBINATIONS. 

382.  Permutations  are  the  different  orders  in  which 
things  can  be  placed ;  as,  ab  and  ba. 

Note. — Observe  that  "  things  "  does  not  mean  quantities.  Letters 
or  figures  which  represent  quantities  may  be  the  things  whose  permuta- 
tions are  considered,  but  they  are  regarded  merely  as  so  many  objects 
which  may  be  arranged  in  different  orders. 

383.  Combinations  are  the  different  groups  that  can 
be  formed  of  any  number  of  things,  taking  a  given  number  at 
a  time. 

Thus,  from  the  letters  abc  we  can  form  three  groups  if  we  take  two 
at  a  time ;  viz.,  ab,  ae,  and  be.  The  order  in  which  the  individuals  are 
placed  in  the  group  is  not  considered,  ab  and  ba  being  but  one 
combination. 

384.  The  object  of  the  theory  of  permutations  and  com- 
binations is  to  determine  the  number  of  orders  in  which 
things  can  be  placed,  and  the  number  of  groups  that  can 
be  formed. 

385.  The  two  problems  may  be  thus  stated : 

I.  To  find  the  number  of  permutations  of  n  things  taken 
m  at  a  time. 

II.  To  find  the  number  of  combinations  of  n  things  taken 
m  at  a  time. 

Note. — For  convenience  we  adopt  the  notation,  P  for  permutations 
and  G  for  combinations,  and  to  indicate  the  number  of  things  and  the 
number  taken  at  a  time,  write  a  subscript  fraction  whose  denominator 
shall  be  the  whole  number  of  things  and  the  numerator  the  number 


164      PERMUTATIONS     AND     COMBINATIONS. 

taken  at  a  time.  Thus,  P5  =  number  of  permutations  of  S  things  taken 
3  at  a  time  ;  and  C±  —  number  of  combinations  of  S  things  taken  3  at  a 
time. 

386.  To  find  a  general  expression  for  Pm  and  Cm  we  have, 

n  n 

evidently,  P\  =  n,  and  if  we  take  2  at  a  time  P2  =  n(n — 1); 

;i  n 

for  each  of  the  n  permutations  formed  by  taking  one  at  a 
time,  may  be  placed  before  each  of  the  (n  —  1)  remaining 

things. 

By  the  same  reasoning  we  shall  have 

P3  =  n  (n  —  1)  (n  —  2),        and 

Pn  —  n  (n  —  1)  (n  —  2)  .  .  .  (n  —  m  +  1),         (1) 

Pn  =  n  (n  —  1)  (n  —  2)  .  .  .  1  =  \n.  (2) 

n 

387.  Every  combination  of  w  things,  taken  mata  time, 
may  have  Pm  =  |m  permutations. 

Therefore,  \m  x  Cm  =  Pm 

n  n 

■L   m 

-        n  (n  —i)....(«  —  ;??  4-  1) 
or  C/m  =  — -  = -■  (3) 

388.  If  the  things  whose  permutations  are  to  be  found  are 
not  all  dissimilar,  the  number  of  permutations  will  be  less, 
being  evidently  divided  by  the  number  of  permutations  which 
could  be  made  with  the  identical  things  if  they  were  unlike 
That  is,  if  p  of  the  things  are  alike  and  q  others  are  alike,  etc., 
then 

\n 

Pl =  Q^lptc7  (4) 


PERMUTATIONS     AND     COMBINATIONS.        165 


EXAMPLES. 

i.  How  many  permutations  can  be  made  with  the  9  digits 
taken  all  at  a  time  ?     That  is,  P9   =  what  ? 

2.  Pr,  =  what?  4.     (74  =  what? 

3.  <%  =  what  ?  5.     Show  that  C»  =  CU, . 

6.  How  many  products  can  be  formed  of  six  factors  taken 
two  at  a  time  ? 

7.  What  is  the  number  of  products  that  can  be  formed 
from  the  9  digits,  taken  3  at  a  time  ? 

8.  How  many  more  products  can  be  formed  of  50  numbers 
taken  40  at  a  time,  than  taken  1  o  at  a  time  ? 

9.  P„  -i-  Cs  =  what  ? 

n  n 

10.  C &  —  C9    =  what? 

11.  If  P5  =  120C3,  what  is  the  value  of  w  ? 

12.  Find  the  value  of  m  that  will  make  Cm.  the  greatest 
possible.  2" 

13.  Find  the  value  of  m  that  will  make  C  m    the  greatest 
possible.  2"+1 

14.  How  many  permutations  (P„)  can  be  made  with  the 

"  Ll2 

letters  of  the  word  Permutations'?  Ans.  -. — 

15.  How  many  permutations  can  be  made  with  the  letters 
of  the  word  Ecclesiastical  ? 

16.  How  many  permutations  can  be  made  with  the  letters 
of  the  word  Divisibility  ? 


CHAPTER    XV. 

INFINITESIMAL    ANALYSIS. 

389.  In  the  preceding  chapters,  quantities  have  been 
distinguished  as  hnoivn  and  unknown.  The  problems  consid- 
ered have  involved  quantities  to  which  arbitrary  values  could 
be  assigned,  and  others  whose  values  were  to  be  found  from 
these.  In  these  problems,  changes  in  the  values  of  quantities 
have  been  made  by  adding  or  subtracting  finite  differences. 

390.  But  there  is  a  large  class  of  problems,  in  which  quan- 
tities must  be  conceived  as  passing  from  one  value  to  another 
by  a  process  of  growth. 

A  quantity  changing  in  this  way  from  one  value  to 
another,  passes  through  all  intermediate  values ;  as,  when  a 
point  moves  from  one  position  to  another,  it  passes  through 
all  intermediate  positions ;  or  as  time  in  passing  from  one 
hour  to  another,  passes  through  every  instant  of  intermediate 
time. 

This  leads  to  the  conception  of  quantities  as  constant  and 
variable. 

391.  A  Variable  is  a  quantity  conceived  as  changing 
from  one  value  to  another  in  such  manner  as  to  pass  through 
all  intermediate  values. 

392.  A  Constant  is  a  quantity  whose  value  remains  the 
same  during  the  same  discussion. 

Constants  are  of  two  kinds,  absolute  and  arbitrary. 

393.  An  Absolute  Constant  is  a  quantity  expressed 
by  a  number  whose  value  never  changes;  as,  4.  10,  etc. 


INFINITESIMAL     ANALYSIS.  107 

394.  An  Arbitrary  Constant  is  represented  by  one 

or  more  letters,  to  which  values  may  be  arbitrarily  assigned, 
but  which  remain  the  same  throughout  the  same  discussion. 

395.  Consecutive  Values  of  a  variable  are  those 
values  between  which  there  are  no  intermediate  values. 

396.  The  Differential  of  a  variable  is  the  difference 
between  two  of  its  consecutive  values. 

Note — It  is  evidently  impossible  to  conceive  of  this  difference  ;  for 
any  conceivable  difference  would  imply  intermediate  values  ;  but  the 
existence  of  strictly  consecutive  values  is  a  necessity  from  the  manner  in 
which  a  variable  changes  its  value. 

397.  A  Function  (in  the  m6nitesimal  analysis)  is  a 
quantity  or  algebraic  expression  whose  value  depends  on  one 
or  more  variables. 

A  function  is  therefore  a  variable  quantity,  having  its 
consecutive  values  corresponding  to  the  consecutive  values  of 
the  variables  on  which  it  depends. 

398.  The  Differential  of  a  function  is  the  difference 
between  two  of  its  consecutive  values. 

399.  We  may  suppose  the  infinitesimal  increments  or 
differentials  by  which  a  variable  passes  from  one  value  to 
another  to  be  equal  to  each  other ;  that  is,  we  may  assume  the 
differential  of  a  variable  to  be  constant,  since  there  is  nothing 
to  forbid  such  a  supposition,  but  the  differential  of  the  func- 
tion which  depends  on  the  differential  of  the  variable  will  not 
usually  be  constant.     Hence, 

400.  The   Differential  of  a    Function  may  be 

defined  as  the  infinitesimal  change  in  the  function  produced  by 
a  change  in  the  variable  from  one  value  to  its  consecutive  value. 

401.  Differentiation  is  the  process  of  finding  a  gen- 
eral expression  for  the  difference  between  any  two  consecutive 
values  of  a  function ;  in  other  words,  of  finding  a  general 
expression  for  the  differential  of  a  function. 


168  INFINITESIMAL     ANALYSIS. 

402.  The  variable  whose  increments  are  arbitrarily  assumed 
is  called  the  Independent  Variable.    (Art.  399.) 

403.  The  function  whose  increments  depend  on  the  incre- 
ments of  the  independent  variable  is  called  the  Dependent 

Variable. 

Thus,  in  the  expression  x°-  —  x,  if  we  assume  that  x  increases  by  a 
constant  increment,  x  will  be  the  independent  and  x-  —  x  the  dependt  id 
variable,  or,  making  u  —  x-  —  x,   u  is  the  dependent  variable. 

404.  Infinitesimal  quantities,  expressed  in  terms  of  any 
finite  unit,  are  all  o.  (Art.  42,  6th.)  In  order,  therefore,  to 
express  the  relations  of  infinitesimals  to  each  other,  fome 
infinitesimal  unit  must  be  employed.  It  need  be  no  objection 
to  the  use  of  such  a  unit,  that  the  conception  of  its  magnitude 
is  impossible,  for  mathematics  is  concerned  only  with  the 
ratios  of  quantities,  and  never  with  their  absolute  magnitude. 
It  is  of  no  advantage  whatever  to  the  mathematician  to  know 
the  magnitude  of  the  unit  employed. 

We  may  therefore  assume  as  the  unit  of  measure  for 
differentials  the  Differential  of  the  Independent 
Variable.     (Art.  399.) 

405.  To  Differentiate  a  Function  will  then  be.  to 
find  the  differential  of  the  function  in  terms  of  the  differen- 
tial of  the  variable  on  which  the  function  depends. 

NOTATION. 

406.  The  Differential  of  a  Variable  is  expressed 
by  writing  before  it  the  letter  d;  thus,  dx,  dy,  dz,  etc.,  to  be 
read,  "differential  of  .r,"  "differential  of//,"  etc. 

Note. — The  d  in  these  expressions  is  not  a  factor,  but  only  an 
abbreviation  of  the  word  differential,  and  must  be  so  read. 

The  same  abbreviation  placed  before  a  function  indicates 
the  differential  of  the  function;  as,  d(x2),  d(a?—x),  are 
read  and  mean,  "the  differential  of  aP/V'the  differential  of 
a:2  —  x." 

In  all  such  cases,  the  function  whose  differential  is  expressed  must 
be  enclosed  in  a  parenthesis.  Thus,  d  {.)■-)  is  the  differential  of  x2 ;  but 
da?  is  the  square  of  the  differential  of  x. 


INFINITESIMAL     ANALYSIS.  1G9 

407.  When  any  function  of  a  variable  is  under  discussion 
to  avoid  its  repetition,  we  write  the  variable  in  parentheses 
with/;  0,  or  ip  before  it;  as,  f(x),  f(ij),  etc.,  read,  "function 
of  x,"  "function  of  y,"  etc. 

Different  functions  arc  expressed  by  fi  (x),  f(x),  <p(x), 
etc.,  read,  "function  prime  of  x,"  "function  sub  one  of  x," 
"  the  <p  function  of  x,"  etc. 

408.  When  either  of  these  symbols  is  used  for  any  func- 
tion, it  represents  the  same  function  throughout  the  same 
discussion. 

409.  The  differential  of  a  function  is  usually  a  variable, 
and  may  therefore  be  differentiated.  Its  differential  is  called 
a  second  differential,  and  is  expressed  thus,  d2u,  d%y,  read, 
"second  differential  of  u,"  etc.  The  meaning  of  these  will  be 
more  fully  explained  hereafter. 

410.  The  Differential  Coefficient  of  a  function  is 
the  ratio  of  the  differential  of  the  function  to  the  differential 
of  the  variable  on  which  the  function  depends.  If  u  represent 
the  function  and  x  the  variable,  the  differential  coefficient  is 

du 

expressed  by  —r-  • 

411.  Infinitesimals  and  infinites  are  of  different  orders, 
depending  on  the  number  of  such  factors  they  contain. 

Thus,  dx,  dy,  dz,  etc.,  are  of  the  first  order,  having  but  one  infinitesi- 
mal factor  ;  dx*,  dy-,  dx  dy,  etc.,  are  of  the  second  order,  and  dx3,  dy3, 
dx2  dy,  dx  dy  dz,  etc.,  are  of  the  third  order,  and  so  on  for  higher  orders. 

So  also  co'2  is  of  the  second,  and  co3  of  the  third  order. 

It  is  evident  that  an  infinitesimal  of  the  second  order  is 
infinitely  less  than  one  of  the  first ;  one  of  the  third  infinitely 
less  than  one  of  the  second,  and  so  on ;  for  each  additional 
infinitesimal  factor  divides  the  quantity  by  infinity  or  multi- 

,.      .    ,       i 
plies  it  by  — 

In  like  manner,   cc2  is  infinitely  greater  than  oo . 


170  INFINITESIMAL     ANALYSIS. 

412.  We  have  seen  that  when  a  quantity  is  measured  with 
a  unit  infinitely  greater  than  itself,  as  when  an  infinitesimal  is 
measured  with  a  finite  unit,  the  measure  is  zero.  (Art.  404.) 
Hence  it  follows  that, 

In  any  polynomial,  a  term  that  is  infinitely  less  than  another 
term  may  be  treated  as  o  and  omitted. 

That  is,  in  an  expression  containing  finite  and  infinitesimal  terms, 
the  infinitesimal  terms  may  be  dropped  ;  and  in  expressions  containing 
only  infinitesimal  terms,  all  higher  orders  of  infinitesimals  may  be 
dropped.     Thus, 

a  ±  dx  =  a; 
dx  ±  dx2  =  dx ; 
dx  ±  dx  dy  =  dx. 
So  also,  00  ±  a  =  00 , 

and  transposing,  00  —  co  =  ±  a.     (Art.  348.) 

DIFFERENTIATION. 

413.  By  the  definition  of  a  differential  (Art.  400),  we  may 
evidently  Differentiate  a  Function  by  the  following 

GENERAL   RULE. 

I.  Add  to  the  variable  the  differential  of  the  variable. 
II.  Develop  and  simplify  the  expression. 
III.  Subtract  the  primitive  state  of  the  function. 

414.  The  application  of  this  rule  to  different  functions 
leads  to  special  rules  by  which  the  process  of  differentiation  is 
abbreviated. 

1.  Differentiate  ax. 

Let  u  =  ax.  (1) 

Adding  dx  to  x,  u  +  du  =  a  (x  +  dx). 

Developing,  u  +  du  =  ax  +  adx. 
Subtracting  (1),  du  =  adx.  Hence, 

Rule  I. —  The  differential  of  the  product  of  a  variable  by  a 
constant  factor  is  the  product  of  the  constant  factor  by  the 
differential  of  the  variable. 


INFINITESIMAL     ANALYSIS.  171 

2.  Differentiate  ax  —  bx  +  c. 

Let  u  =  ax  —  bx  +  c.  (i) 

Adding  dx  to  x,  u  +  du  =  a  (x  +  dx)  —  b  (x  +  dx)  +  c. 

Developing,  u  +  du  =.  ax  +  adx  —  bx  —  bdx  +  c. 
Subtracting  (i),  du  =  adx  —  bdx.  Hence, 

Rule  II. — Hie  differential  of  a  polynomial  is  the  algebraic 
sum  of  the  differentials  of  its  several  terms,  the  differential  of 
a  constant  term  being  zero. 

3.  Differentiate  vyz,  in  which  v,  y,  and  z  represent  func- 
tions of  x. 

Let  u  =  vyz.  (1) 

Adding  dx  to  #,  u  +  du  =  (v  +  dv)  (y  +  dy)  (z  +  dz). 

Developing,  etc.,  u  +  du  =  vyz  +  vydz  +  vzdy  +  yzdv. 
Subtracting  (1),  du  =  vydz  +  vzdy  +  yzdv.  Hence, 

Rule  III. — The  differential  of  the  product  of  several  func- 
tions is  the  sum  of  the  products  of  the  differential  of  each 
function  by  the  product  of  the  other  functions. 

Notes. — 1.  Observe  that  as  v,  y,  and  z,  are  functions  of  x,  adding  dx 
to  x  adds  to  each  function  the  differential  of  that  function,  that  is,  dv  to 
v,  dy  to  y,  and  dz  to  2. 

2.  In  reducing  the  expression  after  it  is  developed,  the  terms 
vdydz  +  ydvdz  +  zdvdy  +  dvdydz  are  dropped,  being  infinitesimals 
of  the  second  and  third  orders. 

3.  It  may  be  shown  that  this  rule  applies  to  any  number  of  factors. 

4.  Differentiate  - ,  y  and  z  being  functions  of  x. 


Let 

«  =  £• 

s 

(I 

Adding  dx  to  x, 

u  +  du  =  y-±$. 

z  +  dz 

Subtracting  (1), 

z  +  dz       z 

Reducing, 

z^-ydz, 
22 

Hence, 

172  INFINITESIMAL     ANALYSIS. 

Kule  IV. — Tlie  differential  of  a  fraction  is  the  denomina- 
tor multiplied  by  lite  differential  of  the  numerator  minus  the 
numerator  multiplied  by  the  differential  of  the  denominator, 
divided  by  the  square  of  the  denominator. 

Notes. — I.  If  the  numerator  be  constant,  its  differential  is  zero  and 

the  expression  becomes   — - — . 

z'2 

2.  If  the  denominator  be  constant;  as  '-  /  =  -  y\  it  may  be  differen- 
tiated by  Rule  I. 

5.  Differentiate  xn. 
1st.  When  n  is  a  positive  integer. 
Let  u  =  x*  =  x  .  x  .  x    .    .     to  n  factors. 

By  Rule  III,        du  =  xn~1dx  +  x^dx  +  &c,  to  n  terms. 
du  =  nxn~'idx. 

2d.  When  n  is  a  positive  fraction. 

Let  n  =  — ,    m  and  s  being  positive  integers 

m 

Then  u  =  r«, 

and  11s  =  x"'. 

»us~ldu  —  mx,H~ldx, 

j                    ,         mx'"-* 
and  du  =  -  dx. 

m 

Substituting  x"  for  u  in  the  denominator, 

mx™—1              mxm~x  ,         m--i7 
du  = dx  =  dx  —  —xa     ax. 

8  \Xs)  8X         s 

3d.  When  n  is  negative,  and  either  integral  or  fractional. 

Let  u  =  x~"  =  — . 

X" 

By  Rule  IV,  du  = : — -  =  —  nx~n~hlx. 

x-n 

Hence,  whatever  be  the  value  of  n, 

Rule  V.  —  TJie  differential  of  any  power  of  a  variable 
whose  exponent  is  constant,  is  the  continued  product  of  the 
exponent,  the  variable  with  its  exponent  diminished  by  one,  and 
the  differential  of  the  variable. 


INFINITESIMAL     ANALYSIS.  173 

415.  The  formulas  of  which  these  rules  are  the  translation 
are  more  convenient  to  memorize  and  use  than  the  rules  them- 
selves.    We  therefore  collect  them  below. 

I.     d  {ax)  =  adx. 

II.     d  {ax  —  lx  +  c)   =  adx  —  bdx. 
III.     d  (xyz)   =  xydz  +  xzdy  +  yzdx. 

IV      d(-)  --  ydx  ~  xdy  - 

V.     d  (x")   =z  nx"~}dx. 

EXERCISES. 

i.  Differentiate   5a;3  —  3Z2  -f  jx  —  6.     (See  I,  II,  and  V.) 

A?is.  (152;2  —  6x  +  7)  dx. 

2.  Differentiate   {x  —  a)  (x  +  b).     (See  III.) 

Ans.  {x  —  a)  dx  +  (x  +  #)  £&c  =  {2X  —  a  +  b)  dx. 

3.  Differentiate (See  IV.) 

—  {a  -\-  x)  dx  —  («  —  x)  dx  —  2a 

AnS.  -.  rz —r— — ■ —r-  dX. 

{a  +  xy  {a  +xf 

4.  Differentiate  y  =  ax%  +  bx  —  c. 

Ans.     dy  =  {2ax  -f-  b)  dx. 

Dividing  by  dx,  we  have        ~  =  2ax  +  b. 

416.  This  is  called  the  first  differential  coefficient  of  the 
function  ax2  +  bx  —  c.  (Art.  410.)  It  is  the  factor  which 
multiplied  by  dx  gives  dy,  or  the  differential  of  the  function. 

This  function  (2ax  +  b)  may  be  differentiated  again,  or 
we  may  differentiate 

dy  =  (2ax  +  b)  dx. 

Remembering  that  dx  is  constant  and  differentiating, 
we  have 

d?y  =  2adx2, 


174  INFINITESIMAL     ANALYSIS. 

which  is  the  second  differential  of  the  function 
ax1  +  bx  —  c. 
Dividing  by  dx?,  we  have 
d~y 
dx2  ~~ 

417.  This  is  called  the  second  differential  coefficient  of  the 
function.  It  is  the  ratio  of  the  second  differential  of  the 
function  to  the  square  of  the  differential  of  the  variable. 
This  function  (2a)  is  constant,  therefore  its  differential  will 
be  zero,  and  the  third  differential  of  ax2  +  bx  —  c  is  zero. 

Differentiate  the  following  and  find  their  successive  differ- 
ential coefficients. 

5.  y  =  ax*  —  bx3  -f  ex  —  ab. 

Ans.    -jf-   =  4ax3  —  ^bx2  +  c. 


d?y 
dx2  ~ 

1 2  ax2  - 

—  6bx. 

d3y 
da?  ~ 

?4ax  - 

-6b. 

d*y 

dx*  ~ 

24a. 

d*y  _ 

ax5 

0. 

6.  y  =  gx5  +  5^  +  3^  —  7^2  —  5- 

Ans.     j-  =  45^  +  2o.f3  +  q.r2  —  14a;. 

d2v 

-~  =  i8oz3  -+-  60a;2  +  iSz  —  14. 

Cb3u 

"  =  540a;2  +  120.?;  4-18. 


dx3 
d*y_ 
dx*  ~ 
<&y_ 

dx5 

dH 

— —  =  o. 

dx6 


10802  +  120. 
1080. 


INFINITESIMAL     ANALYSIS.  175 

7.    y  =  {x  —  a)  (x  —  b)  (x  —  c). 

x%  +  1 


y  = 


9-  y  = 


X  +   I 

a* 


z2  4-  1 

10.  y  =  £*  —  gx*  —  53^  +  7. 

1 

11.  y  = 

*        1  —  a; 

12.  y  =  A  +  5*  +  Cr2  4-  Dx3. 
J3-    2/  =  0  +  a;)5- 

Considering  the  binomial  1  +  x  the  variable,  we  have 

<?y  =  5  (1  +  a;)4  d  (1  +  a;)  =  5  (1  +  a;)4  efo.     (Art.  415,  V.) 

14-  y  =  (1  —  ^)4- 

15-  y  =  (*  +  *)*. 

16.   ?/  =  (1  —  »)m. 

17.  y  =  z7  —  5^  +  3^a  +  5- 

18.  y  =  [x  —  1)  (x  4-  2)  (a;  —  5)  (z  4-  3). 


19-  y 


=  </* 


/ 


1  4-  X 


z=r,  the  differential  of  which  is 

1  +  x         y  I  +  a- 

-* * i+  *  — ~ ■     (Art.  415,  IV.) 

mi       c  j     .  j  a;2  +  2a;  +  a2         , 

Therefore,  reducing,  rfy  = 1 5  »#• 

2  (a2  —  a:2)?  (1  +  x)z 

./ax 

21.  ?/  =  (1  4-  x)~^. 

22.  y  =1  (1  —  a:2)"2. 

23.  _?/  =  Vfl  4-  ^2- 

24.  w  =  a%3  4-  xsy2. 

25.  »  =  1  j3  —  i.r2  4-  5. 


CHAPTER    XVI. 

INDETERMINATE    COEFFICIENTS. 

418.  Indeterminate  Coefficients  are  letters  assumed 
to  represent  certain  unknown  coefficients  during  the  process 
by  which  these  coefficients  are  determined. 

419.  The  Theory  of  Indeterminate  Coefficients 

is  expressed  by  the  following 

Theorem. 

If  a  'polynomial  function  of  a  single  variable  be  equal  to 
zero  for  all  values  of  that  variable,  the  coefficients  of  the  differ- 
ent potvers  of  the  variable  ivill  each  be  equal  to  zero. 

Demonstration. — Let 

Axa  +  Bxb  +  Cxc  +  Bx&  +  etc.  =  o  (i) 

be  the  given  equation,  in  which  a  <  b  <  c,  etc.,  and  a  is  positive.  This 
involves  no  contradiction  of  the  hypothesis,  for  the  terms  of  the  equation 
may  be  arranged  in  any  order,  and  the  equation  may  be  cleared  of  frac- 
tions without  affecting  its  truth.     Dividing  this  equation  by  w  gives 

A  +  Bxt*-11  +  Cxc~a  +  Bxd~a  +  etc.  =  o. 

Since  this  equation  is  true  for  all  values  of  x,  it  is  true  when  x  r=  o, 
a  supposition  which  gives  A  =  o.  The  term  .1./"  may  therefore  be 
dropped,  and  the  resulting  equation  divided  by  xh,  giving 

B  +  Cxc-b  +  Bxd~b  +  etc.  =  o. 

Making  x  =  o  gives  B  =  o.  In  like  manner,  we  may  show  that  all 
the  coefficients  are  zero. 

Cor. — If  an  equation  between  two  polynomial  functions  of* 

a  simile  variable  be  true  for  all  values  of  that  variable,  the 
equation  will  be  identical. 


INDETERMINATE     COEFFICIENTS.  177 

For,  transposing  all  the  terms  to  one  member,  the  coefficients  will  all 
become  zero,  which  would  not  be  the  case  if  the  equation  were  not 
identical ;  that  is,  the  coefficients  of  the  like  powers  of  the  variable  will 
be  the  same  in  the  two  members. 

420.  The  most  important  applications  of  the  Theory  of 
Indeterminate  Coefficients  are, 

i.    To  the  development  of  functions. 

2.  To  the  decomposition  of  fractions. 

3.  To  the  reversion  of  series. 

421.  A  function  is  developed  when  some  indicated 
operation  is  performed. 


EXAMPLES. 
1.  Develop  •■ 

OPERATION. 

Assume   =  A  +  Bx  +  Cx*  +  Dxz  +.Exi  +  etc.  (1) 

1  +  x  w 

Clearing  of  fractions  and  transposing  the  terms  to  the  second  member, 

0=      A  \  +  A  \x  +  B  \x2  +  C  \x3  +  D  ^x*  +  E  \x5  -t  etc.     (2) 

-  i\  +  B\      +  C\       +  B\      +  E\      +  F\ 

Since  x  is  a  variable,  equation  (1)  is  true  for  all  values  of  x,  and  we 
have  from  Equation  (2),  by  the  Theorem, 

A  —  I  =  o ;     A  +  B  —  o  ;     B  +  C  =  o ;     0+Z>  =  o;     etc. 
A  =  1;  B  =  —  1 ;  C=i;  B  =  —  1 ;    etc. 

Note. — The  values  of  these  coefficients  enable  us  to  determine  the 
law  of  the  series,  and  to  write  any  required  number  of  terms,  as  follows  : 

1 
=  1—  x  +  x-  —  x3  +  x*  —  xs  +  etc. 

I  +  X 


178  INDETERMINATE     COEFFICIENTS, 

2.  Develop  (a  —  xp. 


OPERATION. 

Assume    (a  —  a;)*  =  A  +  Bx  +  C'X!  +  Dx3  +  Ex4  +  etc.  (i) 

Squaring, 

a  —  x  =  A*  +  lABx  +  lAC    x-  +  2AD  j  x3  +  zAE\  x4  +  etc. 
+  B-  +  2BC  I       +  2BD  j 

+  C- 
.'.    Art.  419,  Cor., 

A2  =  a ;  A  =  a}  ; 

2AB  =  —  1 :  B  = 


1 


2a 
1 

1 

5 
128a7 
etc.  etc. 


2AG  +  B*  =  o;  G=- 

2AD  +  2BG  =  o ;  ,  D  =  - 

2AE  +  2BD  +  C*  =  o ;  E  =  - 


Substituting  in  (1), 

,1         1       x         a;2         x3  sx4 

(a  —  x)*  =  a*  —  — k - - — - ,    etc. 

2a3      8a11      16a3      128a5 

422.  From  these  solutions  we  have  for  the  development 
of  a  function  of  a  single  variable  the  following 

Rule. — I.  Assume  the  function  equal  to  a  series  with 
indeterminate  coefficients,  containing  all  the  fmvers  of  the 
variable  which  the  development  requires. 

II.  Free  this  equation  from  fraction*  and  from  parentheses 
which  include  different  powers  of  the  variable,  and  make  the 
coefficients  of  like  powers  of  the  variable  in  the  two  members 
equal  to  each  other  ;  or  transpose  all  the  terms  to  one  member, 
and  make  the  several  coefficients  equal  to  zero. 

III.  From  the  equal  inns  /has  formed  find  the  values  of  the 
coefficients  and  substitute  them  in  the  assumed  series. 

Notes. — 1.  The  form  of  the  function  must  determine  what  powers 
of  the  variable  to  assume.  One  thing  should  always  be  observed,  viz.  . 
The  assumed  equation  should  give  no  absurd  result  when  x  —  o.  Thus, 
if  iu   equation   (1),  Example    1,   x  =  o,   A  =  I,  a  result  involving  no 


INDETERMINATE     COEFFICIENTS.  179 

absurdity;   but  if  the  series  were  Ax  +  Bxi  +  Cxz,  etc.,  x  =  o  would 
give  i  =  o,  an  absurdity. 

2.  If  any  power  or  powers  of  the  variable  belonging  to  the  develop- 
ment are  omitted  from  the  assumed  series,  it  will  be  shown  by  some 
such  absurdity  in  the  course  of  the  solution,  if  it  does  not  appear  by 
making  x  =  o. 

3.  If  powers  of  the  variable  not  found  in  the  development  be 
assumed,  their  coefficients  will  be  found  to  be  zero,  and  the  function  will 
be  correctly  developed.  It  appears,  therefore,  that  it  is  only  necessary 
to  make  sure  of  including  all  necessary  powers,  since  it  does  not  vitiate 
the  development  to  include  those  that  are  unnecessary. 

Let  the  student  illustrate  this  by  assuming  for  the  function  above, 
=  A  +  Ex*  +  CiA  +  Dx*  +  etc. 


1  +  x 


Also,  =  Ax-'2  +  Bx-1  +  C  +  Dx  +  Ex11  +  etc. 


Develop  the  following : 


3- 

1  +  X 

4- 

1 

X  -\-  X2 

5- 

(,-  -  l)i 

6. 

ax  —  x2 

Xs  —  X* 

7. 

(a  +  x)-k 

8. 

a 

(x  +  a)2 

x1  —  1 

9- 

X3  +  X2  —  2X  +    l 

10. 

X  +    I 
I  —  X2 

11. 

(!-*)» 

12.     (a  -f-  x)~\ 

X2  —  2X  +   1 


1o- 

Xs  —  X2 

14. 

a  (a  —  x)~\ 

15- 

(x  -  l)l 

16. 

1  —  yx 

x  —  \/x 

17- 

X  +  \/% 

X2  —  X*   +  X* 

18. 

1 
X  —  X  » 

(1  +  ^)3 

19. 

I 

1 " 
I  —  £« 

1 
X* 

X3 


(1  4-  x*)  (1  —  X?) 


Note. — Observe  what  powers  of  x  will  be  found  iu  the  development 
of  the  last  five  examples. 


180  DECOMPOSITION 


DECOMPOSITION     OF     FRACTIONS. 

423.  A  Rational  Fraction,  a  function  of  a  single 
variable,  whose  denominator  lias  rational  factors,  may  be 
separated  into  two  or  more  fractions,  whose  sum  shall  be 
equal  to  the  given  fraction. 

This  is  called  decomposing  the  fraction,  and  the  several 
fractions  are  called,  partial  fractions. 

424.  The  fraction  to  be  decomposed  is  understood  to  have 
its  numerator  of  a  lower  degree  than  its  denominator ;  other- 
wise it  would  give  by  division  one  or  more  integral  terms. 

425.  The  Decomposition  of  a  Fraction  is  per- 
formed by  the  following 

Bulk — I.  Assume  the  given  fraction  equal  to  the  sum  of 
several  fractions  with  indeterminate  numerators,  and   whose 

denominators  include  all  the  denominators  possible  for  the 
partial  fractions. 

II.  Clear  the  equation  of  fractions,  and  collect  like  powers 
of  the  variable  in  one  term. 

III.  Equate  flic  coefficients  of  these  like  powers,  and  from  the 
equations  thus  formed  determine  the  values  of  the  numerators. 

IV.  Substitute  these  values  in  the  assumed  fr  act  inns. 

426.  The  manner  in  which  the  numerators  and  denomi- 
nators of  the  partial  fractions  are  assumed  needs  special  notica 
By  the  rule,  we  are  to  include  in  the  denominators  all 
denominators  possible  to  the  partial  fractions. 

To  ascertain  what  these  will  be,  it  should  be  observed  that, 
since  the  given  fraction  is  the  sum  of  the  partial  fractions, 
each  denominator  must  be  a  factor  of  the  given  denominator. 
(Art.  197.) 

Tiie  denominator  of  the  given  fraction  being  a  rational 
function  of  a  single  variable,  the  prime  rational  factors  will 
have  one  of  the  following  forms. 

x,     x  ±  a,     or    x2  ±  ax  -\-  b.    (Art.  549,  Cor.  2.) 


OF     FRACTIONS.  181 

Any  one  or  more  of  these  forms  may  be  found  with  any 
exponent,  so  that 

xn,     (x  ±  a)n,     and     (z2  ±  ax  +  b)n, 

will  represent  all  the  different  factors  of  the  given  denom- 
inator. ■** 

427.  Considering  the  first  of  these,  xn,  we  see  that  it  must 
be  one  of  the  partial  denominators,  otherwise  it  would  not  be 
a  factor  of  the  denominator  of  the  sum.     (Art.  197.) 

So  also  xn~x  may  be  one  denominator,  and  in  like  manner 
a?1'2,  xn~3,  ....  x,  are  all  possible  denominators.  Hence 
they  must  all  be  used,  and  from  xn  we  shall  have  the  partial 
denominators  x,  x2,  x3,  .  .  .  .  x11. 

In  like  manner,  (x  ±  a)n  will  give  the  denominators 

x±a,     {x±af,    ....     (z±a)n, 

and  (x2  +  ax  +  b)n  will  give 

x*±ax  +  b,     (x*  ±ax  +  b)%,    ....    {xi±ax  +  b)n. 

428.  To  determine  what  numerator  to  assume  for  each 
denominator,  consider  first  the  proper  form  of  the  numerator 
for  xn.     This  numerator  must  be  independent  of  x,  for  if  it 

(Ax  4-  B\ 
as  —  1,  it  would   form   a  fraction 

capable  of  further  decomposition,  and   the  partial   fractions 
would  not  be  the  simplest  possible.     The  assumed  fraction 

therefore  must  be  of  the  form  — ■ 

xn 

The  same  may  be  said  of  all  the  fractions  with  monomial 

denominators. 

429.  The  numerator  also  for  (x  —  a)n  must  be  independ- 
ent of  x  for  the  same  reason,  and  therefore  all  the  fractions 
having  denominators  of  that  form  will  have  numerators  of  the 
zero  degree. 


182  DECOMPOSITION 

430.  The  denominators  of  the  form  (x~  ±  ax  +  b)n  are 
quadratic  factors,  whose  binomial  factors  are  imaginary,  and 
therefore  cannot  be  further  resolved.  Their  numerators  may 
therefore  contain  the  first  as  well  as  the  zero  power  of  x,  and 
must  be  in  the  form  Ax  +  B. 

This  will  give  for  the  different  forms  of  the  partial  fractions, 

A  B  Cx  +  D 

and 


&'         (x  ±  a)n'  (x2  ±  ax  +  b)n 

In  the  third  form,  a  may  be  zero,  reducing  it  to 

Cx  +  D 

(.r2  +  by 

The  process  will  be  made  plainer  by  the  solution  of  a  few 


i.  Decompose  -= 


EXAMPLES. 

x5  —  3Z3  +  i 


XZ  (X  —  2)3  (a*  +   2)2  (2?  —  2X  +   2)2 
Solution. — Assuming  the  partial  fraction  as  above  indicated, 

;.5_3>C3+I  ABC  D  E 

+  -i  +  —-  +  r~-^  + 


X*  (X—2)3(X-  +  2y  (x'2—2X+2)i        X  0?'2        X—2        (X—2)'!        (X—2,3 

Fx+G      Hx  +  I         Kx  +  L  Mx  +  N 

+    -5—7-   +  7-S—C,   +    ..       _,.-    + 


.{'-'  +  2  (.r  +  2)-  .!■  —  235  +  2  (,C2  — 2.r+2)2 

Note. — This  example  is  given  for  tire  purpose  of  including  all  possi- 
ble forms  of  partial  fractions,  and  the  student  should  compare  it  with  the 
preceding  explanations.  The  complete  solution  would  occupy  too  much 
space  to  be  conveniently  printed.  The  student  may  complete  it  for  his 
own  practice  and  satisfaction. 

_.  .T-  —   2X  +    2 

2.  Decompose  -= 


x3  +  2.1-2  —  x  —  2 

Solution. — The  factors  of  the  denominator  are  x+i,  x—i,  and  x+2. 
Assume 

x"  —  2.r  +  2  A  B  G 

X3  +   2J.'2  —  X  —  2  _  X  +    I         X  —   X        X+2 


OF     FRACTIONS 


Clearing  of  fractions  and  uniting  terms, 


183 


X2  —  2X  +   2  =   —  2A 
+   2B 


+    A  I  x  +  A  I  & 
+  3B\     +  b 


c  \  +  u 

.-.      -  2^   +   25   -   C  =  2. 

J.  +  3#  =  -  2. 
A  +  5  +  G  =  i. 

From  which,  -A  =  —  s,  •*»  —  * » 

i  io 

a;-'  —  2*  +  2       _        5 , .  +  —- — — -  • 

•••     *  +  *=*=*-  ~  2(x  +  D  +  6(*  -  i)       3.<*  +  ») 

3-     Decompose    ^T^ZT^) 
4.     Decompose    -—-^-^ 


5.  Decompose    prZl^+T)  (a?  +  s  +  4) 

/^4    _t_   ^;2   _1_    J 

6.  Decompose     ^^T^TTt?' 

a;3  —  3a;  -f  3 

7.  Decompose     -— — ^^—^^ 

«#  —  a2 

8.  Decompose    -nr-^T+V 

1 

9.  Decompose       .  __  ^ 

10.     Decompose     ^r^+7^1 


11.  Decompose     &  +  ^  _  bx  _  ab 

2X5   +   3iC4  _   7^3  _|_   9a;2  _  (,x  +   4 

12.  Decompose (^T^)^-  t) 


CHAPTER    XVII. 

DEMONSTRATION     OF     THE     BINOMIAL 
THEOREM. 

431.  The  Binomial  Formula  has  already  been 
given,  and  the  student  is  familiar  with  its  use.  It  only 
remains  to  give  the  proof,  which  he  was  not  prepared  to 
understand  at  an  earlier  stage  of  his  progress. 

432.  Let  it  be  required  to  develop  (a  +  x)n  into  a  series 
of  ascending  powers  of  x,  n  being  any  number  whatever, 
positive  or  negative,  integral  or  fractional. 

(a  +  .r)"  —  ah  1 1  +  - 1  . 

Put  -  =  g, 

a 

Then        n"(i  +  *)    =  an  (i  +  z)n, 

and  the  development  of  (i  +  z)n  will  give  the  required  series  when  the 
value  of  z  is  restored  aud  the  series  multiplied  by  a".     Assume, 

(i  +  z)«  =  A  +  Bz  +  Cz*  +  Bz3  +  Ez*  +  etc.,  (i) 

in  which  A,  B,  C,  etc. ,  are  indeterminate  coefficients,  whose  values  are  to 
be  found. 

Performing-  the  successive  differentiations  of  (i)  and  dividing  each 
by  dz,  we  have 

n  (i  +2)»-i  =  B  +  2Cz  +  ?,Dz*  +  \Ez3  +  $Fz4  +  etc.  (2) 

n  (1,  - 1)  (1  +g)"-2  =  2C  +  2  .  3DZ  +  3  •  4Ezi  +  4  ■  5F23  +  etc.  (3) 

n(n  — 1)(»  —  2)(i+2)»-3  =  2 -3D  +  2.  3.4^3  +  3-4-5F22  +  etc.  (4) 
n(n—  i)(n-2)(n-3)(i+z)"-i  =  2 .  3  .  4E  +  2  •  3  4.  5^2  +  etc.  (5) 
n(n—i){n—2)(n  —  3)  (n  -  4)  {1  +  z)»-5  =  2.3.4-  5-^+  etc.  (6) 


BINOMIAL      THEOREM.  185 

Making  z  =  o,  we  have  from  these  equations 

A  =  i  ;  B  —  n; 

G_  n(n—i) 


I) 


I  •  2 
_  71  (ft  —  I)  (71  —  2) 


I-  2-3 

£"  —  n(n—  i)(n  —  2)  (ft  —  3)  _ 
1.2.3.4 

^,  _  7^  (ft  -  I)  (ft  —  2)  (ft  —  3)  (ft  -  4) 
I.2-3-4-5 

from  which  we  readily  determine  the  law  of  the  coefficients,  and  may 
write  the  series, 

,         \  n{n—i)„      n(n  —  i)(?i—  2)  , 

(1  +  z)»  -  i  +  nz  +  — ys2  +  — —    -— '  z3  +  etc. 

1-2  1-2-3 

Restoring  the  value  of  a  and  multiplying  by  an,  we  have  the  formula  as 
given  in  Art.  268, 

.  .  ft  (ft  —  1)         „  „       71  (ft  —  1)  (ft  —  2)         „  , 

(a  +  x)»  =  an  +  nan~xx  +  — '-a»-2x2  +  -± - '■  a*-SaP 

1-2  1-23 

7t(ft  —   I)(»  —  2)  (ft  —  3)  .     . 

H 1 -- ao-W  +  etc. 

1-2-3-4 

433.   The  mth  term  of  the  series  is 

n(» -i)(»- 2) (»  -m+J) an_m+la;m_1 

I  •  2  •  3  •  4    ....    (7ft  —  I) 

The  (7ft  +  i)th  term  is 

ft  (ft  -  1)  (ft  -  2)  .  .  .  .  (ft  -  7ft  +  2)  (ft  -  7ft  +  1)  a)l_ma;m> 

1-2-34    ■    ■    •    •    (7ft  —  I)  7ft 

Dividing  the  (/ft  +  i)th  term  by  the  m'h  term  gives 

n  —  m  +  1      x       In  +1         \  x 


_i      x  _  /ft  +  I         \ 
a  ~  \    7ft  / 


a 


This  is  the  variable  factor  by  which  any  term  may  be  multiplied 
to  produce  the  next,  7ft  representing  the  number  of  the  term  multiplied. 


186 


BINOMIAL     THEOBEM. 


EXAMPLES 

i.  Expand  by  the  formula, 

2.  Expand  by  the  formula, 

3.  Expand  by  the  formula, 

4.  Expand  by  the  formula, 

5.  Expand  by  the  formula, 

6.  Expand  by  the  formula, 

7.  Expand  by  the  formula, 

8.  Expand  by  the  formula, 

9.  Expand  by  the  formula, 


10.  Expand  by  the  formula, 

11.  Expand  by  the  formula, 

12.  Expand  by  the  formula, 

13.  Expand  by  the  formula, 

14.  Expand  by  the  formula, 

15.  Expand  by  the  formula, 

16.  Expand  by  the  formula, 

17.  Expand  by  the  formula, 

18.  Find  the  11th  term  of  the  development  of  (ax  +  b)K 

19.  Find  the  mth  term  of  the  development  of  (m  —  x)"\ 

20.  Find  the  nP  term  of  the  development  of  (a  —  x)   - 


a  +  b)5. 
a  -  b)i 
a  —  b)-3. 
a  +  b)~\ 
x  -  y)~k 


1 

(x 

-y? 

(m 

+  n)l 

1 

(m 

—  7i)i 

1 

m  +  n)* 
x  —  2  p. 

X  +   2)3. 
X  +   2)i 

ax  +  by)5, 
ax  +  ty)h 
ax  +  %)_? 
Sa  —  7.r)2. 
3«  +  4ar)i 


Note. — The  student  will  observe  that  any  function  of  .?•  which  can 
be  developed  into  a  series  of  ascending  powers  of  .;•,  may  be  developed  by 
the  same  method.  The  general  formula  for  this  is  called,  from  its 
originator,  McLaurin's  formula.     (See  p.  306,  Note.) 


CHAPTER    XVIII. 

LOGARITHMS. 

434.  The  Logarithm  of  a  number  is  the  measure  of  its 
factors. 

In  other  words,  it  is  the  exponent  which  shows  how  many 
times  the  number  contains  a  given  number  as  a  factor. 

435.  Heretofore,  when  we  have  spoken  of  the  measure  of 
a  quantity,  the  measure  of  its  terms  has  been  meant. 

If  we  wish  to  find  the  measure  of  a  line  27  feet  long,  with  the  yard 
as  a  unit,  we  may  do  it  in  either  of  three  ways : 

1st.  Subtract  3  feet  (the  length  of  the  unit)  from  27,  and  again  from 
the  remainder,  and  continue  so  to  do  until  nothing  is  left.  The  number 
of  subtractions  will  be  the  measure  of  the  terms  of  27. 

2d.  Add  3  +  3  +  3,  etc.,  until  the  sum  is  27,  and  the  number  of  times  3 
is  used  will  be  the  same  measure. 

3d.  Divide  27  by  3,  and  the  quotient  will  be  the  measure. 

We  may  represent  the  three  processes  as  follows  : 

1st.     27-3  —  3  —  3  —  3-3  —  3  —  3-3  —  3=    o. 
2d.  3  +  3  +  3  +  3  +  3  +  3  +  3  +  3  +  3  =  27. 

3d.  27  -J-  3  =    9. 

Therefore  9  is  the  measure  of  27  as  a  term  when  3  is  taken  as  the 
unit  term  ;  that  is,  3  taken  9  times  as  a  term  equals  27. 

We  express  this  by  using  9  as  a  coefficient,  thus  9  ■  3,  or  if  we  let  y 
represent  the  yard, 

Then,  gy  =  27  ft. 

436.  In  a  similar  manner  we  may  measure  the  fac- 
tors of  a  number.  The  measure  of  the  terms  of  a  number 
is  the  measure  of  its  effect  when  added  or  subtracted,  while 


188  LOGARITHMS. 

the  measure  of  its  factors  measures  its  effect  when   used  as 
a  multiplier  or  divisor. 

To  find  this  measure  we  must  assume  a  unit  factor,  as  in  measuring 
the  terms  we  assumed  a  unit  term. 

Take  the  same  number  27  to  measure  its  factors,  and  assume  3  as 
the  unit  factor.  As  before,  we  may  find  the  required  measure  in  three 
ways : 

1st.  Take  the  factor  3  from  27,  and  again  from  the  remaining  factors, 
and  so  on  until  no  factor  remains.  The  number  of  times  we  can  remove 
the  factor  3  from  27  will  be  the  measure  required. 

2d.  Use  3  as  a  multiplier  until  the  product  equals  27 ;  and  the  num. 
ber  of  times  3  is  used  will  be  the  measure. 

3d.  By  a  process  to  be  explained  hereafter,  this  measure  can  be 
found. 

The  first  two  processes  we  may  represent  thus : 
1st.  [  (27  h-  3)  h-  3]  ■*■  3  =  1. 
2d.  3  x  3  x  3  —  27. 

In  the  first  case  we  have  taken  the  factor  3  from  27  three  times  and 
1  only  is  left,  which  is  a  factor  of  no  power,  and  occupies  the  same  place 
with  reference  to  factors  that  zero  does  to  terms.     (Art.  152.) 

In  the  second  case  we  find  that  3  used  three  times  as  a  factor 
equals  27. 

Both  give  us  3  as  the  measure  of  the  factors  of  27  with  3  as  the 
unit  factor. 

This  is  expressed  by  using  an  exponent  thus,  33  =  27. 

437.  The  measure  of  the /floors  of  a  quantity  is  expressed 
by  an  Exponent;  the  measure  of  terms  by  a  Coefficient. 

438.  When  in  any  mathematical  computation  only  one 
unit  term  is  employed,  the  symbol  for  that  term  is  omitted,  and 
the  coefficient  or  measure  only  is  written. 

Thus,  the  surveyor  who  measures  all  his  distances  with  the  rod  as  a 
unit,  does  not  write  10  r,  15  r,  20  r,  etc.,  but  simply  10,  15,  20,  etc. 
Yet,  in  his  computations  he  bears  in  mind  the  fact  that  the  symbol  for 
rod  has  been  omitted,  for  when  he  multiplies  two  of  these  measures 
together,  as  15  x  10,  he  calls  it  150  ?•'-,  or  150  square  rods. 


2'   —      2 

26  =        64 

2°       =     I 

22  =     4 

21  =      128 

2-'=    i 

23  =      8 

28  =      256 

2"2   =     J 

24  =   16 

29  =      512 

o-3  —    X 

—      8 

2;'  =  32 

210=   IO24 

r>— 4     1 

2        —  T6" 

LOGARITHMS.  189 

439.  In  a  similar  manner,  computations  in  which  numbers 
are  used  as  factors  may  be  abbreviated  by  measuring  the 
factors  of  each  with  the  same  unit  factor,  and  omitting  that 
factor,  writing  only  the  exponents  which  express  the  measures 
of  the  factors.  Exponents  used  in  this  way,  as  we  have  seen 
by  the  definition,  are  called  Logarithms.     (Art.  434.) 

Thus,  if  we  adopt  2  as  the  unit  factor,  we  have  2  as  the  measure  of 
the  factors  of  4,  3  as  the  measure  of  the  factors  of  8,  and  4  of  16,  aud  so 
on.     That  is,  we  have  : 

2-5=  A 

2~6  =     «V 

2        —   T2¥ 

c— 6   _      1 
2  —    256 

o—9   —       1 
2         —    51? 

Note. — The  abbreviation  "  log."  is  used  to  express  logarithms.  Thus, 
log.  16  indicates  the  logarithm  of  16,  and  is  read  "logarithm  of  16." 

440.  The  factor  adopted  as  the  unit  factor  is  called  the 
Base.     This  may  be  any  number  except  1. 

That  1  cannot  be  used  as  the  unit  factor  in  measuring  the  factors  of 
other  quantities  is  evident,  since  1  as  a  factor  has  no  power,  and  it  would 
require  an  infinite  number  of  such  factors  to  produce  any  number 
greater  than  1. 

For  the  same  reason  we  cannot  use  zero  for  the  unit  of  measure  for 
terms. 

441.  In  measuring  the  factors  of  numbers  in  this  manner, 
the  signs  of  the  numbers  are  not  and  cannot  be  considered. 
We  measure  only  the  factors  of  magnitude  or  value,  and  not 
the  factors  of  direction.  Hence  the  base  or  unit  of  measure 
for  these  factors  is  a  number  without  a  sign. 

442.  But  the  factors  measured  may  be  used  as  multipliers 
or  divisors,  and  the  measure  should  indicate  this.  As  the 
measure  of  terms  is  either  +  or  — ,  according  as  the  terms  are 
used  in  addition  or  subtraction,  so  the  measure  of  factors 
(logarithms)  are  either  +  or  — ,  according  as  the  factors  are 
used  as  multipliers  or  divisors. 


190 


LOGARITHMS, 


For  example,  if  we  measure  the  factors  of  8  with  2  as  a  unit  factor, 
the  measure  is  +3  ;  but  if  the  8  be  used  as  a  divisor,  making  it  -J-,  the 
measure  is  —3. 

This  will  be  further  illustrated  by  referring  to  Art.  439.  We  have 
from  that,  when  2  is  the  base, 

log.  2  =  1  log.  1  —      o 

"4  =  2  "  £  =  -1 

"8  =  3  -  i  =  -2 

"    16  =  4  «  1  =  _3 


1  — 


-4 


If  now  we  make  2~'  (=  I)  the  base,  we  have, 


log.    2   =    —  I 

log.  1  = 

O 

"       4  =    -2 

«     1  

I 

"     8  =  -3 

•<     1  — 

2 

"   16  =  —4 

"       1    

8     — 

3,  etc.,  etc 

If  we  take  the  base  5,  we  have, 

log.    1=0 

log.  I     = 

0 

"      5  =  i 

"      A    - 

5        — 

—  1 

"    25  =  2 

•'           1        — 
^5      — 

—  2 

"  125  =  3 

«           1        

3 

If  5-1  (=  1)  be  taken  as 

the  base, 

then 

log.    1  =      0 

log.   I    = 

0 

"      5  =  -1 

"        I    = 

1 

"    25  =  —2 

"        1     =: 

2 

"  125  =  -3 

"     jh  = 

3 

443.  We  see  that  log.  1  will  evidently  be  o  for  all  bases, 
since  1  as  a  factor  has  no  power,  and  its  measure  with  any  unit 
factor  must  be  o.  So  also  the  logarithm  of  the  base  will 
always  be  1,  since  the  measure  of  any  quantity  with  itself  as 
a  unit  must  be  1. 

444.  The  logarithms  given  above  are  all  integral,  because 
we  have  selected  such  numbers  as  contained  the  base  as  a 
factor  an  integral  number  of  times. 

Intermediate  numbers  have  intermediate  logarithms. 


LOGARITHMS 


191 


This  may  be  illustrated  by  taking  16  as  a  base.     We  shall  then  have, 

log:,  i   =      o 


Og.    I 

= 

0 

"       2 

= 

1 

4 

= 

•25 

"     4 

= 

1 

= 

•5 

"     8 

= 

3 

= 

•75 

"   16 

= 

I 

"   32 

= 

5 

= 

1-25 

"  64 

= 

3 

= 

i-5 

'<     1 

= 

1 

i 

= 

- 

•25 

<<     1 

= 

1 

5 

= 

— 

■5 

"■     1 

8 

= 

3 

= 

— 

•75 

16 

= 

—  I 

"        1 
32 

= 

5 

3" 

= 

—  I 

•25 

<<         1 
6? 

= 

3 

2 

= 

—  I 

5 

The  logarithms  of  numbers  between  these  are  incommensurable,  and 
can  be  expressed  only  by  approximation. 

445.  The  logarithms  of  numbers  with  a  given  base  are 
called  a  system  of  logarithms.* 

The  system  in  common  use  has  10  for  a  base,  and  from  its 
originator  is  called  Briggs'  System.    In  this  system, 


log. 


1=0 
10  =  1 

100  =  2 
1000  =  3 


The  logarithms  of  numbers 


between 


and 


10 
100 


.1 
.01 


10 
100 
1000 
.1 
.01 
.001 


log.  .1  =  -1 
"  .01  =  —2 
"  .001  =  -3 
"  .0001  =  —4 

s    o  +  a  decimal. 

1  + 

2  + 

—  1  +  " 

—  2  + 
-3  + 


446.  The  integral  part  of  a  logarithm  is  called  its  CJiftv- 
acteristic,  and  the  decimal  part  its  Mantissa.  It  is 
customary  in  expressing  negative  logarithms  to  make  the 
mantissa  positive,  as  indicated  above. 

Thus,  if  we  have  the  logarithm  —2.754526  (the  whole  being  nega- 
tive), it  may  be  expressed  thus, 

-  3  +  -245474, 
or,  as  it  is  commonly  written, 

3.245474, 

in  which  the  3  only  is  negative,  the  sign  —  being  placed  over  it  to  indi- 
cate this. 

*  The  method  of  computing  by  logarithms  was  invented  by  Lord  Napier.  (See 
p.  306,  Notes  3  and  4.) 


192  L0GAKIT1IMS. 

447.  It  will  readily  appear  that  the  characteristic  of  the 
logarithm  of  a  number  in  the  common  system  may  be  known 
by  the  following 

Eule. —  The  number  of  places  from  the  first  significant 
figure  of  a  number  to  units'  place,  counting  the  latter,  is  equal 
to  the  characteristic  of  the  common  logarithm  of  thai  number. 
When  counted  to  the  right,  the  characteristic  is  positive;  when 
counted  to  the  left,  negative. 

Note. — The  reason  fortius  rule  is  found  in  the  fact  that  multiplying 
or  dividing  a  number  by  10  moves  the  decimal  point  one  place  to  the 
right  or  left,  and  at  the  same  time  increases  or  diminishes  its  logarithm 
one  unit.  Hence,  moving  the  decimal  point  of  a  number  to  the  right  or 
left  does  not  affect  the  mantissa  of  its  logarithm. 

448.  What  are  the  characteristics  of  the  logarithms  of  the 
following  numbers: 


I. 

2785.62. 

Ans.  3. 

2. 

5437L 

Ans.  4. 

3- 

.0075. 

Ans.  —  3. 

4- 

1.075. 

5- 

16.0005. 

6. 

.0000589. 

10.     5.89. 

7- 

0.00589. 

11.     589. 

8. 

0.0589. 

12.     58900. 

9- 

0.5S9. 

13.     5890000 

449.  When  the  number  has  integral  figures, 

The  characteristic  of  its  logarithm  is  one  less  than  the 
number  of  integral  places,  and  is  positive. 

450.  When  the  number  is  entirely  decimal. 

The  characteristic  of  its  logarithm  is  one  more  than  the 
number  of  ciphers  between  the  decimal  point  and  the  first 
significant  figure,  and  is  negative. 


LOGARITHMS.  193 


TABLES     OF     LOGARITHMS. 

451.  A  Table  of  Logarithms  is  one  which  contains 
the  logarithms  of  all  numbers  between  given  limits. 

452.  The  Table  found  on  the  following  pages  gives  the 
mantissas  of  common  logarithms  to  five  decimal  places  for 
all  numbers  from  i  to  iooo,  inclusive. 

The  characteristics  are  omitted,  and  must  be  supplied  by 
inspection.     (Arts.  447-450.) 

Notes. — 1.  The  first  decimal  figure  in  column  0  is  often  the  same 
for  several  successive  numbers,  but  is  printed  only  once,  and  is  under- 
stood to  belong  to  each  of  the  blank  places  below  it. 

2.  The  character  (  ♦ )  shows  that  the  figure  belonging  to  the  place  it 
occupies  has  change!  from  9  to  o,  and  through  the  rest  of  this  line  the 
first  figure  of  the  mantissa  stands  in  the  nest  line  below. 

453.  To  Find  the  Logarithm  of  any  Number  from  I  to  10. 

Rule. — Look  for  the  given  number  in  the  first  line  of  the 
table  ;  its  logarithm  will  be  found  directly  below  it. 

1.  Find  the  logarithm  of  7.  Ans.  0.84510. 

2.  Find  the  logarithm  of  9.  Ans.  0.95424. 

454.  To  Find  the  Logarithm  of  any  Number  from  10  to 

1000,  inclusive. 

Rule. — Look  in  the  column  marked  A7"  for  the  first  tivo 
figures  of  the  given  number,  and  for  the  third  at  the  head 
of  one  of  the  other  columns. 

Under  this  third  figure,  and  opposite  the  first  two,  ivill 
be  found  the  last  four  decimal  figures  of  the  logarithm.  The 
first  one  is  found  in  the  column  marked  0. 

To  this  decimal  prefix  the  proper  characteristic.   (Art.  447.) 

Note. — If  the  number  has  4  or  more  figures,  find  the  logarithm  of 
the  first  three  figures  and  add  to  it  the  product  of  the  remaininrr  figures 
considered  as  a  decimal,  by  the  tabular  difference  (from  column  D)  oppo- 
site the  logarithm  of  the  first  three  figures. 


194  LOGARITHMS. 

3.  Find  the  logarithm  of  108.  Ans.  2.03342. 

4.  Find  the  logarithm  of  176.  Ans.  2.24551. 

5.  Find  the  logarithm  of  1999.  Ans.  3.30085. 

455.  To  Find  the  Logarithm  of  a  Decimal  Fraction. 

Rule. —  Take  out  the  logarithm  of  a  whole  number  consist- 
ing of  the  same  figures,  and  prefix  to  it  the  proper  negative 
characteristic.     (Art.  450.) 

Note. — If  the  number  consist  of  an  integer  and  a  decimal,  find  the 
logarithm  in  the  same  manner  as  if  all  the  figures  were  integers,  and 
prefix  the  characteristic  which  belongs  to  the  integral  part.     (Art.  449.) 

6.  What  is  the  log.  of  0.95  ?  Ans.  1.97772. 

7.  What  is  the  log.  of  0.0125  ?  Ans.  2.09691. 

8.  What  is  the  log.  of  0.0075  '•  Ans.  3.87506. 

9.  What  is  the  log.  of  16.45  ?  Ans.  1.21616. 
10.  What  is  the  log.  of  185.3  •  Ans.  2.26787. 

456.  To  Find  the  Number  belonging  to  a  given  Logarithm. 

Rule. — I.  Find  in  the  table  the  mantissa  next  less  than  the 
mantissa  of  the  given  logarithm,  and  the  corresponding  number 
will  be  the  first  three  figures  of  the  required  number. 

II.  Subtract  the  mantissa  found  in  the  table  from  the 
mantissa  of  the  given  logarithm,  and  divide  the  remainder 
by  the  corresponding  tabular  difference,  for  the  remaining 
figures  of  the  required  number. 

III.  Place  the  decimal  point  of  this  number  as  required 
by  the  characteristic  of  the  given  logarithm.     (Art.  447.) 

Note. — If  the  characteristic  of  a  logarithm  be  negative,  the  number 
belonging  to  it  is  a  fraction,  and  as  many  ciphers  must  be  prefixed  to 
the  number  found  in  the  table,  as  there  are  itnits  in  the  characteristic 
leas  1.    (Art.  450.) 


LOGARITHMS.  195 

ii.  What  number  belongs  to  2. 17231  ?  Ans.  148.7. 

12.  What  number  belongs  to  1. 25261  ?  Ans.  17.89. 

13.  What  number  belongs  to  3.27715  ?  Ans.  1893. 

14.  What  number  belongs  to  2.30963  P  Ans.  204. 

15.  What  number  belongs  to  4.29797  ?  Ans.  19858.29. 

16.  What  number  belongs  to  1. 14488?  A ns.  0.1396. 

17.  What  number  belongs  to  2.29136  ?  Ans.  0.01956. 

18.  What  number  belongs  to  3.30928  ?  Ans.  0.002038. 

457.  Computations  in  which  numbers  are  used  as  factors 
may  be  made  by  logarithms  upon  the  following  principles: 

i°.  TJie  sum  of  the  logarithms  of  two  numbers  is  equal  to 
the  logarithm  of  their  product. 

Let  a  and  c  denote  any  two  numbers,  m  and  n  their  logarithms,  and 
6  the  base. 

Then  bm  =  a 

And  bn  =  c 

Multiplying,  &'«+"  =  etc. 

20.  TJie  logarithm  of  the  dividend  diminished  by  the 
logarithm  of  the  divisor  is  equal  to  the  logarithm  of  the 
quotient. 

Let  a  and  c  denote  any  two  numbers,  m  and  n  their  logarithms,  and 
b  the  base. 

Then  b"1  =  a 

And  b"  =  c 

Dividing,  6m—"  =  a  -~  c. 

458.  To  Multiply  Numbers  by  their  Logarithms. 

Eule. —  Add  the  logarithms  of  the  factors  ;  the  sum  will 
be  the  logarithm  of  the  product.     (Art.  457,  i°.) 

Notes. — 1.  If  one  or  more  of  the  characteristics  be  negative,  make 
them  positive  by  adding  10  to  each,  and  reject  as  many  io's  from  the  sum. 
•     2.  The  sign  of  the  product  or  quotient  is  determined  as  when  multi- 
plication and  division  are  performed  in  tbe  usual  manner.     (Art.  129.) 


196  LOGARITHMS. 

i.  Required  the  product  of  35  by  23. 
Solution. — The  log.  of  35  =  1.54407 

"     "   23  =  1. 36173 

Adding,  2.90580. 

The  corresponding  number  is  805,  Ans. 

2.  What  is  the  product  of  109.3  by  14.17  ? 

3.  What  is  the  product  of  — 1.465  by  —1.347  ? 

4.  What  is  the  product  of  .074  by  —1500? 

459.  To  Divide  by  Logarithms. 

Rule. — From  the  logarithm  of  the  dividend  subtract  the 
logarithm  of  the  divisor  ;  the  difference  will  be. the  logarithm 
of  the  quotient.     (Art,  457,  20.) 

Note. — If  either  or  both  characteristics  are  negative,  add  10  to  each 
and  the  result  will  not  be  affected. 

5.  Required  the  quotient  of  120  by  15. 
Solution. — The  log.  of  120  =  2.07918 

"     "    15  =  1. 17609 

"       "      "  quotient  =  0.90309.     Ans.  8. 

6.  What  is  the  quotient  of  12.48  by  0.16  ? 

7.  What  is  the  quotient  of  .045  by  1.20  ? 

8.  What  is  the  quotient  of  1.381  by  .096  ? 

9.  Divide   — 128   by    — 47. 

10.  Divide    — 186   by    — 0.064. 

11.  Divide    — 0.156   by    —0.86. 

12.  Divide    — 0.194  by  0.042. 

460.  To  Involve  a  Number  by  Logarithms. 

Rule. — Multiply  the  logarithm  of  the  number  by  the 
exponent  of  the  required  power. 

Notr. — 1.  This  rule  depends  upon  the  principle  that  logarithms  are 
the  exponents  of  powers,  and  a  power  is  involved  by  multiplying  its 
exponent  into  the  exponent  of  the  required  power. 

2.  Let  the  student  remember  that  power  includes  also  root.  (Art.  253.) 


LOGARITHMS.  197 

13.  What  is  the  cube  of  1.246  ? 

Solution. — The  log.  of  1.246  is  0.09551 

Index  of  the  required  power  i3  3 

Logarithm  of  power  is  0.28653.     Ans.  1.93435. 

14.  What  is  the  fourth  power  of  .  135  ? 

15.  What  is  the  tenth  power  of  1.42  ? 

16.  What  is  the  twenty-fifth  power  of  1.234  ? 

17.  What  is  the  square  root  of  1.69  ? 

Note. — Log.  1.69  must  he  multiplied  by  4.  or  what  is  the  same 
thing,  divided  by  2. 

18.  What  is  the  cube  root  of  143.2  ? 

19.  What  is  the  sixth  root  of  1.62  ? 

20.  What  is  the  eighth  root  of  1549  ? 
2i.  What  is  the  tenth  root  of  1876  ? 

461.  If  the  characteristic,  of  the  logarithm  be  negative, 
and  caunot  be  divided  by  the  index  of  the  required  root 
without  a  remainder,  add  to  it  such  a  negative  number  as  will 
make  it  exactly  divisible  by  the  divisor,  and  prefix  an  equal 
positive  number  to  the  decimal  part  of  the  logarithm. 

22.  It  is  required  to  find  the  cube  root  of  .0164. 

Solution. — The  log  of  .0164  is  2.21484. 

Preparing  the  log.,       3)3+  1.2 1484 

140494.    Ans.  0.25406  +  . 

23.  What  is  the  sixth  root  of  .001624  ? 

24.  What  is  the  seventh  root  of  .01449  ? 

25.  What  is  the  eighth  root  of  .0001236  ? 


198 


LOGARITHMS 


TABLE 

O  F 

COMMON      LOGARITHMS. 


N. 

0 

1 

.00000 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

.3oio3 

•477 '2 

.60206 

.69897 

.778.5 

.845io 

.90309 

.95424 

10 

.ooooo 

0432 

0860 

1284 

1703 

2119 

253 1 

2938 

3342 

3743 

416 

11 

4i39 

4532 

4922 

53o8 

5690 

6070 

6446 

6819 

7188 

7555 

379 

12 

7918 

8279 

8636 

8991 

9342 

9691 

♦037 

♦38o 

♦721 

1059 

349 

i3 

.11394 

1727 

2057 

2385 

2710  ! 

3o33 

3354 

3672 

3988 

43oi 

322 

14 

46i3 

4922 

5229 

5534 

5836 

6137 

6435 

6732 

7026 

7319 

3oi 

i5 

7609 

7898 

8184 

8469 

8752 

9o33 

9312 

9590 

9866 

♦  140 

281 

16 

.20412 

o683 

0952 

1219 

1484 

1748 

201 1 

2272 

253 1 

2789 

264 

n 

3o45 

33oo 

3553 

38o5 

4o55 

43o4 

455 1   4797 

5o42 

5285 

249 

18 

5527 

5768 

6007 

6245 

6482 

6717 

6951 

7184 

7416 

7646 

235 

19 

7875 

8io3 

833o 

8556 

8780 

9003 

9226 

9447 

9667 

9885 

223 

20 

.3oio3 

0320 

o535 

0750 

0963 

1 175 

1387 

1 597 

1806 

2oi5 

212 

21 

2222 

2428 

2634 

2838 

3o4i 

3244 

3445  3646 

3846 

4044 

203 

22 

4242 

4439 

4635 

483o 

5o25 

52i8 

341 1   56o3 

5793 
7658 

5984 

ig3 

23 

6i73 

636 1 

6549 

6736 

6922 

7107 

-291   7473 

7840 

1 85 

24 

8021 

8202 

8382 

856i 

8739 

8917 

9094 

9270 

9445 

9620 

.78 

25 

9794 

9967 

♦  140 

♦3 1 2 

♦483 

♦654 

♦824 

♦993 

1162 

i33o 

•7i 

26 

}.4i497 

1664 

i83o 

1996 

2160 

2325 

2488 

265i 

28i3 

2975 

1 65 

27 

3i36 

3297 

3457 

36i6 

3775 

3933 

,'"/! 

4248 

4404 

456o 

1 58 

28 

4716 

4871 

5o25 

5'79 

5332 

5485 

5637  : 

5939 

6090 

1 53 

29 

6240 

638g 

6538 

6687 

6835 

6982 

7129 

7276 

7422 

7567 

i47 

3o 

•477 '2 

7857 

8001 

8i44 

8287 

843o 

8572 

8714 

8855 

8996 

i43 

3i 

91 36 

9276 

941 5 

9554 

9693 

983 1 

9969 

♦  106 

♦  243 

♦379 

i38 

32 

.5o5i5 

o65i 

0786 

0920 

io55 

1188 

l322 

1455 

1 587 

1720 

i33 

33 

i85i 

i983 

2114 

2244 

2375 

25o4 

2634 

2763 

2892 

3020 

i3o 

34 

3 148 

3275 

34o3 

3529 

3656 

3782 

3908 

4o33 

41 58 

4283 

126 

35 

4407 

453 1 

4654 

4777 

4900 

5o23 

5 1 45 

5267 

5388 

55o9 

123 

36 

563o 

5751 

5871 

599i 

61 10 

6229 

6348 

6467 

6585 

6703 

119 

37 

6820 

6937 

7024 

7'7i 

7287 

74o3 

7519 

7634 

7749 

7864 

116 

38 

7978 

8o93 

8206 

8320 

8433 

8546 

8659 

8771 

8883 

8995 

n3 

39 

9106 

9218 

9329 

9439 

955o 

9660 

977° 

9879 

9988 

♦097 

1 10 

4o 

.60206 

o3i4 

0423 

o53i 

o638 

0746 

o853 

0959 

1066 

1172 

108 

4i 

1278 

i38  i  ',   1490 

i5g5 

1700 

i8o5 

1909 

2014  2118 

2221 

io5 

42 

2325 

2428 

253 1 

2634 

2737 

2839 

2941 

3o43  3 144 

3246 

102 

43 

3347 

3448 

3548 

3649 

3749 

3849 

3949 

4048  ,<  r 

4246 

100 

44 

4345 

4444 

4542 

4640 

4738 

4933  5o3i   5 128 

5225 

98 

45 

5321 

5418 

55 14 

56io 

5706 

58oi 

58o6 

5992  6  >s- 

6181 

95 

46 

6276 

6370 

6464 

6558 

6652 

6745 

6839 

6g32 

7117 

93 

8 

7210 

7302 

7394 

7486 

7578 
8485 

7669 

7761 

-  ■ 

7943 

8o34 

V 

8124 

82i5 

83o5 

83q5 

8574 

8664 

8753 

8842 

8o3i 

89 

49 

9020 

9108 

9'97 

9285 

9373 

946i 

1 

9548 
6 

9636 

9723 

9810 

88 

N. 

0 

1 

2 

3 

4 

5 

7 

8 

9 

D. 

L  0  G  A  K  I  T  II  _M  S  . 


199 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

♦672 

D. 

5o 

.69897 

9984 

♦070 

♦  1 57 

♦  243 

♦329 

♦4i5 

♦5oi 

♦586 

86 

5i 

.70757 

0842 

0927 

1012 

1096 

1181 

1265 

1 349 

1433 

l5l7 

85 

52 

1600 

1684 

1767 

i85o 

1933 

2016 

2099 

2181 

2263 

2346 

83 

53 

2428 

2  509 

2591 

2673 

2754 

2835 

2916 

2997 

3078 

3 159 

81 

54 

3239 

3320 

3400 

3480 

356o 

3640 

3719 

3799 

3878 

3957 

80 

55 

.74o36 

41 1 5 

4194 

4273 

435 1 

4429 

4507 

4586 

4663 

4741 

78 

56 

4819 

4896 

4974 

5o5i 

5i28 

52o5 

5282 

5358 

5435 

55u 

77 

57 

5587 

5664 

5740 

58i5 

589i 

5967 

6042 

6118 

6193 

6268 

76 

58 

6343 

6418 

6492 

6567 

6641 

6716 

6790 

6864 

6g38 

7012 

75 

59 

7o85 

7i59 

7232 

73o5 

7379 

7432 

7525 

7597 

7670 

7743 

73 

6o 

.778i5 

7887 

7960 

8o32 

8104 

8176 

8247 

83i9 

83go 

8462 

72 

6i 

8533 

8604 

8675 

8746 

8817 

8888 

8958 

9029 

9099 

9169 

7' 

62 

9239 

9309 

9379 

9449 

95l9 

9588 

9657 

9727 

9796 

9865 

69 

63 

9934 

♦oo3 

♦072 

♦  140 

♦  209 

♦  277 

♦346 

♦4i4 

♦482 

♦55o 

68 

64 

.80618 

0686 

0754 

0821 

0889 

0956 

1023 

1090 

11 58 

1224 

67 

65 

1291 

i358 

1425 

1491 

1 558 

1624 

1690 

1757 

1823 

1889 

66 

66 

1934 

2020 

2086 

2l5j 

2217 

2282 

2347 

24i3 

2478 

2543 

65 

67 

2607 

2672 

2737 

2802 

2866 

2930 

2995 

3o59 

3i23 

3187 

64 

68 

32DI 

33t5 

3378 

3442 

35o6 

3569 

3632 

3696 

3759 

3822 

63 

69 

3885 

3948 

401 1 

4073 

4i36 

4198 

4261 

4323 

4386 

4448 

63 

7° 

.845io 

4572 

4634 

4696 

4757 

4819 

4880 

4942 

5oo3 

5o65 

62 

7i 

5i  26 

5i87 

5248 

5309 

5370 

543 1 

5491 

5552 

56i2 

5673 

61 

72 

5733 

5794 

5854 

5914 

5974 

6o34 

6094 

6i53 

62i3 

6273 

60 

73 

6332 

6392 

645i 

65io 

6370 

6629 

6688 

6747 

6806 

6864 

59 

74 

6923 

6982 

7040 

7099 

7i57 

7216 

7274 

7332 

739o 

7448 

52 

75 

75o6 

7364 

7622 

7680 

7737 

7795 

7852 

7910 

7067 

8024 

58 

76 

8081 

8i38 

8196 

8252 

83o9 

8366 

8423 

8480 

8536 

85o3 

57 

77 

8649 

8705 

8762 

8818 

8874 

8930 

8986 

9042 

9098 

9134 

56 

78 

9209 

9265 

9321 

9376 

9432 

9487 

9542 

9597 

g653 

9708 

55 

79 

9763 

9818 

9873 

9927 

9982 

♦037 

♦091 

♦  146 

♦200 

♦  255 

55 

80 

.90309 

o363 

0417 

0472 

o526 

o58o 

o634 

0687 

0741 

0795 

54 

81 

0849 

0902 

0936 

1009 

1062 

1 1 16 

1 169 

1222 

1275 

i328 

54 

82 

i38i 

1434 

1487 

1 540 

i593 

i645 

1698 

i75i 

i8o3 

i855 

32 

83 

1908 

i960 

2012 

2o65 

21 17 

2169 

2221 

2273 

2324 

2376 

52 

84 

2428 

2480 

253i 

2583 

2634 

2686 

2737 

2788 

2840 

2891 

52 

85 

2942 

2993 

3o44 

3095 

3i46 

3197 

3247 

3298 

3349 

3399 

5i 

86 

345o 

3  5oo 

355i 

3  60 1 

365 1 

3702 

3752 

38o2 

3852 

3go2 

5i 

87 

3952 

4002 

4o52 

4101 

41 5i 

4201 

425o 

43oo 

435o 

4399 

30 

88 

4448 

4498 

4547 

4596 

4645 

4694 

4743 

4792 

4841 

4890 

49 

89 

4939 

4988 

5o36 

5o85 

5i34 

5i82 

523i 

5279 

5328 

5376 

48 

90 

.95424 

5472 

5321 

5569 

56i7 

5665 

57i3 

576i 

5809 

5856 

48 

91 

5904 

5932 

6000 

6047 

6095 

6142 

6190 

6237 

6284 

6332 

47 

92 

6379 

6426 

6473 

6320 

6567 

6614 

6661 

6708 

6755 

6802 

47 

93 

6848 

6893 

6942 

6988 

7o35 

7081 

7128 

7174 

7220 

7267 

46 

94 

73 1 3 

7359 

7403 

745i 

7497 

7543 

7589 

7635 

7681 

7727 

46 

95 

7772 

7818 

7864 

7909 

7935 

8000 

8046 

8091 

8137 

8182 

45 

96 

8227 

8272 

83i8 

8363 

8408 

8453 

8498 

8543 

8588 

8632 

45 

97 

8677 

8722 

8767 

881 1 

8856 

8900 

8945 

8989 

9034 

9078 

45 

98 

9123 

9167 

921 1 

9255 

93oo 

9344 

9388 

9432 

9476 

9520 

44 

99 

9364 

9607 

9631 
2 

9695 
3 

9739 

9782 

9826  9870 

99'3 

9957 

43 

N. 

0 

1 

4 

5 

6 

7 

8 

9 

D. 

200  LOGARITHMS. 

COMPUTATION     OF     LOGARITHMS. 

Theorem  I. 

462.  In  any  two  systems  of  logarithms,  the  logarithms  of 
like  numbers  have  a  constant  ratio. 

Demonstration. — Let  the  bases  of  the  systems  be  a  and  a',  and  let 
x  be  any  number  whose  logarithms  in  the  two  systems  are  s  and  z '. 
That  is, 

az  =  x  and  a'~   =  x. 

(i) 


az  =  a!* 

Let 

am  —  a'. 

Substituting  in  (i), 

az   —   awz' 

.•.     z  = 

mz 

or 

z 

-  =  m, 

2 

But  since  a  and  a'  are  constant,  in  is  also  constant. 

Cor.  i.  —  The  logarithm  of  a  number  consists  of  two  factors, 
one  of  which  is  a  function  of  the  base  and  the  other  a  function 
of  the  number. 

This  is  evident,  since  changing  the  base  introduces  or  removes  a 
constant  factor  and  makes  no  other  change  in  the  logarithm. 

463.  The  Modulus  of  a  System  is  this  constant 
factor,  depending  on  the  base,  and  is  usually  represented  by  M. 

The  logarithm  of  x  will  therefore  be  Mf{x),  in  which  M 

is  a  function  of  the  base  of  the  system  and  is  therefore  constant. 

Cor.  2. — TJie  base  may  be  so  chosen  as  to  malce  the  modulus  i . 

For  in  the  equation  am  =  a',  it  is  evident  that  a'  may  be  so  taken  as 
to  give  m  any  value  whatever;  m  may  therefore  be  made  such  as  to 
cancel  the  factor  M,  or  modulus  ;  so  that,  in  z  —  mz',  the  constant  factor 
of  2' shall  be  1. 

464.  Baron  Napier,  who  first  suggested  this  use  of  expo- 
nents, took  such  a  base  for  his  system  of  logarithms,  and  they 
are  called  from  the  inventor  the  Napierian  system. 


LOGARITHMS.  201 

The  Napierian  Logarithm  of  z  will  therefore  be 
simply  f(x),  the  modulus  beiug  i. 

Note. — The  Napierian  system  of  logarithms  is  sometimes  called  the 
natural  system,  on  account  of  its  relation  to  other  systems.  Napierian 
logarithms  are  also  called  hyperbolic,  logarithms,  by  reason  of  their  rela- 
tions to  certain  areas  connected  with  the  hyperbola.  The  base  of  this 
system,  commonly  represented  by  e,  is  2.718281  +  .     (Art.  468.) 

465.  Logarithms  of  different  systems  may  be  expressed  by 
writing  the  base  of  the  system  subscript  to  the  abbreviation 
log. 

Thus,  loge  indicates  a  Napierian,  and  logio  a  common  logarithm. 

When  no  subscript  figure  or  letter  is  used,  the  abbreviation 
log.  must  be  understood  to  mean  the  common  logarithm. 

We  may  also  write  M\o,  Me,  etc.,  for  the  moduli  of  the  different 
systems. 

Coe.  3. — Loga  x  =  Ma  log6  x ;  Ma  being  the  modulus  of 
the  system  whose  base  is  a. 

For        log(l  x  =  M„f(x)    and    logex=f(x).    (Arts.  463-464.) 
It  follows  therefore  that 

466.  The  modulus  of  any  system  is  the  ratio  of  any 
logarithm  in  that  system  to  the  Napierian  logarithm  of  the 
same  number.     Hence, 

467.  A  table  of  logarithms  with  any  base  may  be  con- 
structed by  multiplying  the  logarithms  of  the  Napierian 
system  by  the  modulus  of  the  required  system. 

Cor.  4. — The  logarithms  of  the  same  number  in  different 
systems  are  to  each  other  as  their  moduli. 

For  loga  X  =  Ma  fix)       and       loga'  X  =  Ma'f(x). 

•      log«  ■*'    _  M±  m 

loga'  X    ~  Ma'' 


202  COMPUTATION     OF 


Theorem    II. 


468.  The  differential  of  the  logarithm  of  a  variable  is 
equal  to  the  modulus  of  the  system  multiplied  by  the  differential 
of  the  variable  divided  by  the  variable. 

Let                                u  =  loga  x.  (i) 

Adding-  dx  to  x,   u  +  du  =  log,,  (x  +  dx). 

Subtracting  (i),            du  =  log„  (i  +  —  \  =  Maf(x  +  — Y    (2) 

(Arts.  457,  2°,  and  463.) 

To  find  /   H — -J,    let  xt  and  x<>  be  any  two  values  of  x. 

We  may  then  write                            x1  =  x.,",  (3) 

And                                               logaxt  =  n  log,  x.,.  (4) 

Differentiating  (3),                              dx1  =  ?ix.,"-ldx2.  (5) 

Differentiating  (4),                 d(loga  x^  =  nd(\og*  ./-,),  (6) 

,  .          .                                            dxt          dx0  ,  . 

(5)  "5-  (3),                                             -^  (7) 

(6)  -*-  (7), 


dxx 
xx 

dxs 
=  n  — -. 

x3 

d{\ogaxx 

d(\0gaX2) 

dxx 

dxs      ' 

x1 

x2 

,       dx 
du  oc  — . 

dx 

du  =  m  — . 

X 

dx 

m  — 

x 

=  ,/,/(, +f). 

m 

=    Ma, 

dx 

X 

=/(-?> 

du 

dx 

=   Ma . 

X 

du 

=  d(\og.x)  =  *j. 

That  is,  (Arts.  377-380),  du  oc  — .     .-.     du  =  m  — .  (8) 

From  (2)  and  (8), 
Hence  (Art.  419,  Cor.), 
And 

Hence  du  =  Ma—.  (9) 

If  a  =  e,  du  =  dQoge  x)  =  —.        (10) 

468'/.    To  find  the  value  of  e,  we  have  from  (2),  substituting  e  for  a, 

,         .       dx         _,  .  ,    /      dx\        1 

ed"  =  1  H .         For  convenience  put  du\=  —     =  -. 

x  \      x  I       n 

i  1 

Then  c    =  1  +     .  (11) 

11 

Raising  (11)  to  the  ?;th  power, 

1      n(n  —  1)   1        n  (n  —  1)  (»  —  2)   1 

e  =  1  +  n  -  +  — ■ .  -5  +  -  •  —.  +  etc. 

n  1-2       «?  1-2-3  n 

Since      =  — ,  an  infinitesimal,  n  =  cc  ,  and 
w        a; 

a  =  1  +  -  +  —  +  -       -+-  ■-  +  etc.    (Art.  412.)    (12) 

I  1-2  1-2-3  I-2-34 

.•.     e  =  2.71828182S459045 +. 


LOGARITHMS.  203 

469.   To  find  a  formula  for  computing  the   Napierian 
logarithms  of  numbers,  assume 

log*  (i  +  x)  =  A  +  Bx  +  Ox~  4-  Bxs  +  Exi  +  etc.       (i) 

Differentiating  successively,  and  removing  factors  common 
to  both  members. 

=  B  +  2  Ox  +  $Bx*  +  4Ex3  +  etc.  (2) 


1  -f  x 

1 


-  =  2O  +  2-3Dx  +  3.4^2  4-  etc.  (3) 

+  (T  +  a.\3  =  3^  +  3  •  4^»  +  etc.  (4) 

=  AE  +  etc.  (5) 


0 

+  «)4 

-    4-z-'      1 

Hence, 

making 

a  =  0, 

From 

(0. 

<t 

(2), 

u 

(3), 
(4), 

a 

(5), 

A  =  o, 
B=i, 
n  —  1 

£    =    i, 

^  =  —  1,  etc. 

Substituting  in  (1),  and  continuing  the  series  by  the  law 
which  becomes  evident, 

/vtf  /y3  sy*±  /y*5  /yG 

log*  (1  +  x)  =  x L4 1 7-  +  etc.       (6) 

23456  w 

470.  Formula  (6)  is  called  the  Logarithmic  Series, 

but  it  is  not  in  a  form  suited  to  the  computation  of  logarithms, 
as  may  be  seen  by  substituting  6  for  x,  which  gives, 

.  .       62       63       6i 

log,  7=6 1 \-  etc., 

2        3        4 

in  which  the  terms  are  increasing,  and  it  is  impossible  to  take 
any  number  of  terms  which  will  give  an  approximate  value  of 
log*  7- 


204  COMPUTATION     OF 

471.  The  following  transformations  will  adapt  this  formula 
to  the  computation  of  the  Napierian  logarithms  of  numbers. 

Putting  —  x  for  x  in  (6), 

/v2  *y3  nA  o*5 

loge  (i  —  x)  =  —x : etc.  (7) 

2345  v" 

Subtracting  (7)  from  (6), 

log,  (1  +  x)  —  log,  (1  -  x)  =    loge  (73^) 

/         x3       x5       x"1  \ 

=  2[x  +  -  +-+  -  +  etc.) 

^357  ' 

In  this  equation,  making  x  = ,  we  have,  by  writing 

log«  ( i  +  z)  —loge  2    for    loge( ),  and  transposing  loggZ, 

loge(i  +z)=loSez+  2(^  +  -T-^8+_L-_  +  etc.) 

.    .     .     (A) 

472.  To  compute  loge  for  the  numbers  1,   2,  3,  4,  etc., 
we  have,  by  Art.  443, 

loge  1  =  o,  (1)' 

and  by  formula  (A),  making  z  =  1, 

loge(l   +  i)  =  loge  2 

=0+2-4 H H ;  +  etc.)         (2) 

\3      3-3"      5*3B      7*37  ' 

Making  2=2, 

log,  3  =  log, ,  +  .  (l  +  i  +  ^5  +  -I,  +  etc.)        (3)' 

loge  4  =  2  log«  2.     (Art.  457.  i°.)  (4)' 

Making  %  =  4, 

loge 5  =  log,  4  +  2  (I  +  ^  +  ^  +  ^T  +  etc.)         (5)' 


LOGARITHMS. 


205 


473.    To  add  a  sufficient  number  of  terms  of  series  (2)'  to 
give  loge  2  to  six  places  of  decimals,  proceed  as  follows : 


3  ) 

2.00000000 

-   1 

— 

.66666667 

9) 

.666666666  - 

1st  term. 

9) 

.07407407  - 

1 

0 

= 

.02469136 

2d  " 

9) 

.00823045  - 

-  5 

= 

.OO164609 

3d  " 

9) 

.00091449 

-  7 

= 

.OOOI3064 

4th  " 

9) 

.00010161 

-  9 

= 

.OOOOII29 

5th  " 

9) 

.00001129  - 

-11 

= 

.OOOOOIO3 

6  th  " 

9) 

.00000125 

-  r3 

= 

.OOOOOOO9 

7  th  " 

.00000014 

-  !5 

l0ge2 

— 

.OOOOOOOI 

8th  " 

.69314718 

Note. — In  this  computation  the  terms  should  he  carried  tiro  plac  s 
of  decimals  farther  than  the  logarithm  is  to  be  used,  to  insure  accuracy 
in  the  last  figure. 


roni  (3)'  in  like  manner  we  have, 

5  )  2.00000000 

25  )  .40000000 

I 

= 

.40000000 

1  st  term 

25  )  .01600000 

3 

= 

•°°533333 

2d  " 

25  )  .00064000  - 

5 

= 

.00012S00 

3d  « 

25  )  .00002560 

7 

= 

.00000366 

4th  " 

.00000102  - 

°ge 

9 

2 

— 

.0000001 1 
.69314718 

5th  « 

1 

1 

Oge 

3 

= 

1. 09861228 

From  (4)'  we  have, 

logg4  =  2  log*  2  =  1.38629436. 

Since  the  sum  of  the  logarithms  of  several  factors  equals 
the  logarithm  of  their  product,  we  need  only  compute  by  (A) 
the  logarithms  of  the  prime  numbers. 


Let  the  student  find  as  above  Jog*  5,  loge  6,  and  \oge  7. 


206  LOGAEITHMS. 


474.    We  have        loga  x  =  Ma  loge  x.     (Art.  465,  Cor.  3.) 
in  which,  if  x  be  made  a, 


log*  a; 


,r        logo  a  I 

*'A?  =  r-   —  =  ! Hence, 

lo-gg  a       \oge  a 

Tlie  modulus  of  any  system  is  the  reciprocal  of  the  Napierian 
logarithm  of  the  base  of  the  system. 

474.  <i.    Substituting  a—  1  for  x  in  (6),  Art.  469,  we  have, 
log,  a  =  a  -  1  -  I  (« - 1  )2  +  1  (a- 1  )3  -  i  (« - 1  )4  +  etc.    (8) 


log*  a       a—  i-i(«-i)2  +  i(a  —  i)3— J(«  —  i)4  +  etc. 

....   (9) 

^d  Jftt  =  -  ~-^^—l- 

9  —  -  +  y  —  -  +  etc. 

234 

We  also  have,  by  substituting  x  for  a  in  (8), 

loge£  =  .r— 1  —  i{x—  i)2  +  f(a;— i)3  — £(»— i)4+etc.     (10) 

Multiplying  (10)  by  (9)  gives, 

-  x-i  -  j(x-i)*  +  j(x-i)3  -j(x-iy  +  etc. 
logax  -  a_I_^(a_I)2  +  |(a_l)3_:i(a_I)4  +  etc; 

Representing  this  numerator  by  f(x),  the  denominator  will 
be  /(«).  and 

l0gair-/(«) 
Let  the  student  show  that 
The  modulus  of  the  common  system,  Mi0,  is.  43429448  +  . 

475.    To  Compute  a  Table  of  Common  Logarithms. 
Multiply  llir  Napierian  logarithms  by  J/,„  =  .43429448  +  . 
(Art.  467./ 


CHA  PTER    XIX. 

SERIES. 

476.  A  Series  is  a  succession  of  quantities  which  in- 
crease or  decrease  in  accordance  with  some  fixed  law.  These 
quantities  are  called  the  Terms  of  the  series.     For  example, 

Suppose  a  quantity  x  beginning  with  the  value  i  to  increase  in  such 
manner  as  to  double  each  second.  If  we  write  the  values  of  x  taken  at 
equal  intervals  of  time,  they  will  form  a  series.  If  taken  at  intervals  of 
one  second,  the  series  will  be 

i,     2,     4,     8,     16,     32,    etc., 

or  if  taken  at  intervals  of  two  seconds, 

1,     4,     16,     64,     etc. 

Again,  suppose  the  side  of  a  square,  beginning  at  o,  to  increase 
uniformly  at  the  rate  of  one  inch  per  second.  If  the  area  of  the  square 
be  taken  each  second,  we  have  the  series, 

i2,     2-,     3-,    4'2,     52,    etc. 

If  once  in  two  seconds,  we  have, 

i2,     3'2,     52,     72>     etc. 

477.  The  Law  of  a  Series  must  therefore  express  two 
things. 

1st.  The  rate  of  increase  of  the  quantity. 
2d.  The  intervals  of  time  at  which  its  values  are  taken  for 
the  terms  of  the  series. 

The  law  of  a  series  is  usually  expressed  in  the  form  of  a  general  rule 
for  the  formation  of  any  term  from  the  preceding  term  or  terms  ;  or  we 
may  have  several  terms  of  a  series  given  from  which  to  determine  the  law. 


208  SERIES. 

478.  Since  there  is  no  limit  to  the  number  of  different 
laws  which  may  govern  the  formation  of  series,  there  will  be 
an  unlimited  variety  of  series.  Our  space  will  only  allow  the 
discussion  of  a  few  of  the  most  important. 

479.  For  convenience  we  number  the  terms  of  a  series 
from  left  to  right,  beginning  with  some  term  which  we  call 
the  first  term ;  but  as  it  is  evident  that  any  series  may  be 
extended  both  ways,  we  shall  not  only  have  terms  numbered 
i,  2,  3,  4,  5,  etc.,  to  the  right,  but  also  those  numbered  o,  — i, 
—  2,  —3,  —  4,  etc.,  to  the  left, 

480.  The  problems  to  be  discussed  relating  to  series  are, 

1st.  Finding  any  required  term  of  a  series. 
2d.    Interpolation  of  terms. 
3d.   Summation  of  series. 
4th.  Reversion  of  series. 

481.  To  find  any  term  of  a  series  requires  a 
formula  which  will  give  the  value  of  the  variable  quantity  at 
any  given  time ;  as  in  the  series  of  the  squares,  to  find  the 
10th  term  is  the  same  thing  as  finding  the  area  of  the  square 
at  the  end  of  10  seconds. 

482.   Interpolation  of  terms  is  the  process  of  finding 
one  or  more  terms  intermediate  between  any  two  terms  of  a 

series. 

Thus,  if  we  find  the  area  of  the  square  (Art,  476)  at  the  end  of  5^ 
seconds,  we  shall  have  a  term  of  the  series  between  the  5th  and  6th,  and 
equidistant  from  each  ;  that  is,  equidistant  in  time,  but  not  in  value. 

483.  The  practical  value  of  this  problem  may  be  illus- 
trated by  supposing  the  altitude  of  the  sun  to  be  known  for 
noon,  and  for  each  hour  after  noon  till  sunset.  These  alti- 
tudes will  form  a  series,  and  if  the  law  can  be  found  we  can 
find  the  sun's  altitude  for  any  intermediate  time,  say  for  2\, 
2\,  and  2|  o'clock.  These  will  form  three  terms  between  the 
third  and  fourth  terms  of  the  series. 


SERIES.  200 

484.  It  is  evident  that  the  formula  for  finding  any 
term  of  a  series  will  apply  to  interpolation,  for  if  we  can  find 
the  value  of  a  variable  at  the  end  of  10  seconds,  we  can  by 
the  same  process  find  it  for  q\  or  of  seconds. 

485.  The  Summation  of  series  is  the  process  of 
finding  the  sum  of  any  number  of  terms  of  a  series.  The 
method  will  of  course  depend  on  the  law  of  the  series. 

486.  A  Converging  Series  is  one  in  which  the  sum 
of  an  infinite  number  of  terms  infinite. 

487.  A  Diverging  Sei'ies  is  one  in  which  the  sum  of 
an  infinite  number  of  terms  is  infinite. 

488.  A  series  is  increasing  or  decreasing  according  as  its 
successive  terms  increase  or  decrease. 


DIFFERENCE     SERIES. 

489.   Take  the  series 

i     .     5     .     15     .     35     .     70     .     126     .     210.         (1) 

By  subtracting  the  first  term  from  the  second,  the  second  from 
the  third,  and  so  on,  we  have  a  series  of  differences  called  the 
first  order  of  differences. 

From  these  differences  another  set  of  differences  may  be 
formed  in  the  same  way,  called  the  second  order  of  differ- 
ences, and  from  these  the  third  order,  and  so  on  till  the 
differences  become  zero. 

The  series  (1)  and  its  several  orders  of  differences  will  be  as 
follows : 

Series,  1         5         15         35         70         126         210 
1st  order  of  diff.         4         10         20         35         56  84 

2d  order  of  diff.  6         10         15         21  28 

3d  order  of  diff.  4567 

4th  order  of  diff.  1  1  1 

5  th  order  of  diff.  o  p 


210  SERIES. 

490.  A  series  which,  like  the  above,  gives  an  order  of 
differences  equal  to  zero,  is  called  a  Difference  Series. 
Not  only  is  series  (i)  a  difference  series,  but  each  set  of 
differences  is  also  a  difference  series. 

The  series  4,  5,  6,  7,  etc.,  having  its  first  differences  con- 
stant (1  .  1  .  1,  etc.),  is  called  a  difference  series  of  the  first 
order  ;  6  .  10  .  15,  etc.,  having  the  second  differences  constant, 
is  of  the  second  order.  So  also  4  .  10  .  20,  etc.,  is  of  the 
third  order,  and  1  .  5  .  15  .  35,  etc.,  of  the  fourth  order. 

It  is  also  evident  that  this  series  may  be  made  the  differ- 
ences of  a  series  of  the  fifth  order,  and  so  on  indefinitely. 

491.  A  difference  series  of  the  first  order,  is  called  an 
JEquidifferent  Series,  because  each  term  is  formed  by 
adding  a  constant  difference  to  the  term  preceding. 

Note. — This  series  is  commonly  called  an  Arithmetical  Series  or 
Progression. 

492.  To  find  formulas  fo*r  the  nth  term  and  the  sum  of 
n  terms  of  a  difference  series  of  any  order,  take  the  series, 

«x  a2  ai  ai  a5  0« 

1st  order  of  differences, 

a.i—ai  a3—a,  ai—a3  a3— a4  a6—aB 

2d  order  of  differences, 

a3 — 2a2+n1       a4-2«3+a3       a-a—2ai  +  ai       a6—2ah+ai 
3d   order  of  differences, 

"4  —  303  +  30-3  — «i     06— 304  +  30J— «j     "«  — 3«„  +  3«4—03 
4th  order  of  differences, 

05—  4'*4  +6a3— 402  +0i       06—40s  +  6  '4  —  4«a  +as 
5  th  order  of  differences, 

0g  —  505  +  IO04_  I0ax  +  502— 0r 

If  we  put  the  first  terms  of  these  successive  orders  of  differences  = 
dlt  d.,,  d3,  etc.,  we  shall  have, 

(7t  =  a»—  ax. 

d.>  —  a3—2a2  +«,. 

d3  =  «4  —  303  +  303— "l- 

d4  =  a5—  4ai  +  6a3—  4ai+a1. 

d.  =  a*—sas  \-ioai—ioai  +  $a9—al. 


SERIES.  211 

In  these  equations  we  find  the  coefficients  are  the  same  as  for  the  n^  power 
of  a  binomial. 

From  this  we  can  write 

,  n(n— i)  n(n—i)(n—2) 

dn  =  a„+i— nan  +  — an-\ — an-2  + 

i-2  1.2.3 

n(n—  1)  {n— 2)(n— 3) 


1.2.3.4 
Reversing  the  order  of  terms, 


an-z  —  etc. 


,  ,  n  (n — 1)  n(n— i)(n— 2) 

dn  =  ±  a1  =f  na2  ±  — ^ a3  =f  — ! f-± '  a,  ±  etc., 

1-2  1-2.3 

in  which  the   upper  signs  will  be   used  when  n  is  even,  and  the  tower 
when  n  is  odtf. 

From  the  values  of  dlt  d2,  etc.,  we  get, 

a2  =  a1  +dx. 

a3  =  aY  +  2dt  +d.,. 

ai    =  #1  +  3^1  +  3^2  +  ^3- 

a5  =  a1+4d1+6d.,+4d3+di. 
And  from  the  law  of  the  coefficients,  which  is  evident, 

/        nj        (w— 1)  («— 2)  ,        (n  —  i)(n— 2){n— 3), 

a„  =  a,  +  («  — 1)^  +  v 'ds  +  - -ds  +  etc.  (A) 

1-2  1  •  2-3 

493.  From  this  formula  any  term  of  a  difference  series  of 
any  order  may  be  found,  when  enough  of  its  terms  are  known 
to  give  the  first  terms  of  the  several  orders  of  differences.  The 
number  of  terms  of  the  formula  used  in  any  case  will  depend 
on  the  order  of  the  series.  Thus  for  series  of  the  first  order 
(Equidifferent  series),  all  the  differences  after  dx  will  be  zero. 
Hence  the  formula  will  become 

an  =  at  ■+  (n  —  1)  d  (A)' 

corresponding  to  the  common  formula  for  the  nth  term  of  an 
Arithmetical  Series. 

Note. — The  subscript  1  is  omitted  from  d  as  unnecessary. 

494.  To  find  the  Sum  of  n  terms  of  the  Difference 
Series, 

form  another  series  of  which  the  given  series  shall  be  the  first 
order  of  differences  ;  thus, 

0,    av    at  -f  a9,    at  +  r/3  4-  n%i     etc. 


212  SERIES. 

It  is  evident  that  the  {n  +  i)^term  of  this  series  is  the  sum  of  n 
terms  of  the  given  series  ;  hence,  if  we  apply  formula  (A)  and  find  the 
(«  +  i)(A  term  of  this  last  series,  \vu  shall  have  the  sum  of  n  terms  of  the 
given  series,  as  required.  To  make  this  application  we  must  make 
in  (A), 

n  =  n  +  i,        al  =  o,        d1  =  alt        d.,  =  du        etc. 
Making  these  substitutions,  we  have,  putting  a„+1  =  8n, 

a                 ,    n(n — i)   .         n  (n  —  i)  (n  —  2)  ,  ,„, 

Sn  =  na1  +  -* }-dx  +  — K- LS >  d    +  etc.        (B) 

1-2  1-2-3 

Note.— S  is  used  for  the  sum  of  a  series,  with  a  subscript  letter  or 
figure  to  indicate  the  number  of  terms  included. 

495.  If  the  series  be  Equidifferent,  this  becomes 

~  n  (n  —  i)  ,  /T>X( 

Sn  =  na1  +  — v—    — '  d,  (B)' 

1.2 

which,  by  substituting  the  value  of  d  from  (A)',  becomes 

Sn  =  a^-^n.  (B)" 

This  is  the  common  formula  for  the  sum  of  an  Arith- 
metical Series. 

496.  The  formula  for  the  n"'  term  of  a  series  is  also  used 
for  interpolation.  (Art.  484.)  In  that  formula  n,  which 
represents  the  number  of  any  term,  may  be  more  properly 
regarded  as  representing  the  time  at  which  the  value  of  the 
variable  is  taken  for  any  term.  Hence  the  formula  for  the 
nth  term  applies  equally  well  when  n  is  fractional  as  when  it 
is  integral. 

If  therefore  it  be  required  to  interpolate  3  terms  between 
the  9th  and  10th  terms  of  a  series,  we  have  only  to  make 
11  =  9I,  9I,  and  9f,  successively  in  the  formula  for  the 
nth  term  of  the  series. 

When  the  terms  are  known  between  which  other  terms 
are  to  be  interpolated,  the  preceding  terms  of  the  series  may 
be  disregarded,  and  these  two  terms  may  be  called  the  first 
and  second  terms  of  the  series.  One  term  to  be  interpolated 
will  be  the  i|  term,  two  will  be  i\  and  if,  three  will  be  i\, 
1 1,  \\,  and  so  on. 


SERIES.  213 

Note. — Terms  thus  interpolated  are  called  Means,  and  the  terms 
between  which  they  are  inserted  are  called  Extremes. 

497.  When  several  equidifferent  means  are  to  be  inserted 
between  two  extremes,  it  may  be  convenient  to  use  a  formula 
for  (I,  obtained  from  Eq.  (A)' ;  thus, 

On  =  «i  +  (n  —  i)  d, 
gives  d  =  ^£-/  ;  (C) 

in  which  a„  and  at  represent  the  two  extremes,  and  n  the 
number  of  terms  in  the  completed  series,  or  tioo  more  than 
the  number  to  be  interpolated. 

Thus  to  insert  4  means  between  5  and  20,  we  have 

20  —  5 
*  =   6^T  =  3" 

Hence  the  series  complete  is 

5     •     8     •     11     .     14    .     17    .     20. 

By  formula  (A),  a  series  may  also  be  carried  backward,  calling  the 
numbers  of  the  terms  from  the  first  term, 

o,        —1,         —2,        —3,        —4,        etc. 

This  may  be  illustrated  by  the  series, 

1     •     4     •     10     •     20     •     35.        Thus, 

—  5  —4—3—2-10  123  4  5  6 

— 10       — 4       — 1         o     o       o       (1       4       10        20       35)      56 
6  3  1        o       o       1        (3       6        10       15)        21 

—  3       —2   —1      o       1        2       (3       4         5)  6 

1  1        1       1       1         1       (1  1)       1 

The  first  line  shows  the  number  of  each  term  of  the  series.  These  terms 
are  formed  by  extending  the  equidifferent  series,  and,  from  that,  forming 
terms  of  the  series  of  the  second  order,  and  then  those  of  the  third.  We 
may  also  find  any  one  of  these  terms  by  formula  (A).     (Art.  492.) 


214 


SERIES. 


i.  Given  the  series  r  .  8  .  27  .  64. 125  .  etc.,  to  find:  1st. 
the  15th  term  ;  2d,  the  sum  of  15  terms  ;  3d,  the  first  of  three 
terms  interpolated  between  the  5th  and  6th:  4th,  the  —  4th 
term;  5th,  the  n,h  term ;  6th,  the  sum  of  n  terms. 

SOLUTIONS. 

1  8  27  64  125  216 

7  19  37  61  91 

12  18  24  30 

6  6  6 

o  o 

1st.  From  formula  (A)  we  have, 

14  -13         14  •  13  •  12  , 

«15  =  1  +  14-  7  +  ^ ?I2  +  -^ — * 6; 

2  2-3 

«!  B    =    1  +  98  +  IO92  +  2184  =   3375. 


2d.  By  formula  (B), 

*5  •  14        15  •  14-  13 


Si  1 


15  + 


7  + 


12  + 


15  •  14  -  13  •  12 


2  2.3  2-3-4 

815  =  15  +  735  +  5460  +  8190  =  14400. 

3d.  The  first  of  three  terms  inserted  between  the  fifth  and  sixth  will 
be  the  5}  term. 

Using  formula  (A),  we  have, 

ai=  I+4i.7+l|jL3iI2+4i13i12|6 

^4  2  2-3 

a5x  =  1  +  29!  +  82^  +  31&  =  i44||- 
4th.  By  the  same  formula  the  —  4th  term  is 


a—i 


„  r+(-5)7+<-5)(-<i>Is+t-3(-'i>(-7>,U 

2  2-3 


a-i  —  1  — 35  +  180— 210  =  —64. 

5th.  The  same  formula  gives  the  11th  term, 

(«-i)(h-2)         (n-i)(7i-2)(n-3) 
an  -  i  +  (»— 1)7  +  - fi2+i -— o, 

1-2  I-2-3 

a„  =  i  +  7(n—i)  +  6(?i—i)(n  —  2)  +  (n—i)(n  —  2)(7i-3)  -  n3. 


SERIES.  215 

6th.  The  sum  of  n  terms  from  formula  (B)  is 

a  _  n  +  nJP-j)  ?  +  n{n-i)(n-2)  ^  +  n  {n-i)  (n-z)  (n-3)  6  . 
2  2-3  2.3-4 

.-.     Sn  =  n  +  ?,n(n—  i)  +  2n(n—  i)(n— 2)  +  \n()i—  i)(n  —  2)  (»— 3) 
~n  (n  +  i)~ 


_  nnn  +  i)y 


EXAMPLES. 

Find  the  nth  term  and  the  sum  of  n  terms  of  the  following 
series,  and  apply  the  formulas  thus  obtained  by  making  n  = 
different  numbers.  Also  interpolate  terms  until  the  formulas 
are  familiar. 


I 

h 

4, 

7> 

10, 

13,     etc. 

2 

3> 

61, 

10 

,     1 3  J,     17,    etc. 

3 

2, 

7, 

12, 

i7< 

etc. 

4. 

2, 

6, 

i33 

23 

,     etc. 

5. 

i? 

2, 

3. 

4, 

5,     etc. 

6. 

i, 

3> 

5- 

7, 

9,     etc. 

7- 

2, 

4, 

6, 

8, 

etc. 

8. 

i, 

3> 

6, 

10, 

etc. 

9- 

A 

22 

32, 

42 

etc. 

10. 

13, 

23, 

33, 

43 

etc. 

1 1. 

1, 

4- 

10, 

2°. 

35,     etc. 

12. 

i, 

5> 

15, 

35, 

70,     126, 

etc. 

13- 
14. 

1, 

What 

6, 
is  t 

21, 

ie  0 

56. 
ten 

126,     252. 
n,  the  —  xst 

462, 
term 

etc. 


term  of  Ex.  11  above? 

15.  Which  of  the  above  series  gives  the  number  of  balls 
that  can  be  piled  in  a  pyramid  whose  base  is  an  equilateral 
triangle  ?  Which  the  number  that  can  be  piled  in  a  pyramid 
whose  base  is  a  square  ? 


•210  SERIES. 

1 6.  How  many  balls  can  be  piled  in  a  triangular  pyram 
having  10  balls  on  eacb  side  of  the  lowest  tier? 

17.  How  many  in    a  quadrangular  pyramid,  having  tl 
same  number  on  each  side  in  the  lowest  tier  ? 

18.  How  many  in  an  oblong  rectangular  pile  20  balls  Ion 
and  5  balls  wide  ? 

19.  Find  the  —  10th  term  of  Ex.  12  above. 

20.  If  a  body  fall  16  feet  in  one  second,  3  times  as  far  th 
next  second,  5  times  as  far  the  third,  and  so  on,  how  far  wil 
if  fall  the  tenth  second?  How  far  in  10  seconds?  How  fa 
in  i\  seconds?  How  far  in  5^  seconds?  How  far  in  1 
seconds  ? 

2 1 .  Find  from 

a„  =  ay  +  {11  —  i)d, 

Q         n  (al  +   an) 
6B  = -, 

the  following  Formulas  for  Equidifferent  Series  : 

«!  =  (In  —  (n  —  i)d.  (1) 

(2) 

(3) 


2S„ 
CL\   =    

ti 

—  a„. 

ti 

(ti  —  i)d 
2 

«i  =  \  ±  V(an  +  W  ~  ^Sn.  (4) 


an  =  «i  +  {>i  —  *)d.  (5) 

aa  =  — "  -  «,.  (6) 

n 

«„=*.  +  <^M.  (7) 

tl  2 

an  =  -  -  ±  V2dSn  +  (av  -Jdf-  (8) 

2 

10 


SERIES. 

__  n  (a,  +  an)^ 

2 

(9) 

&  =  -  [?ax  +  (»  —  i)<?]. 

(10) 

a„  +  a,      a„2  —  «i2 

0»    =                           +                 7 
2                          2« 

(«) 

&    =    ~  [2tf„  —  (W  —    l)(/J. 

(12) 

7               #H    #1 

«  = 

)l  —  I 

(13) 

_         a„2  —  a,2 

(14) 

2#„  —  an  —  a. 

2  (nan  —  Sn) 

(15) 

n  (n  —  1 ) 

w  (m  —  r) 

(16) 

n  =  -^  +  1. 

(i7) 

n  = • 

(18) 

217 


"1  +  a„ 


d  —  2cix  +  V(2C<i  —  d)*-\-8dS„        ,     , 

"=•-  —2d-'  ~  (I9) 


2a„  +  f?  ±  V(2a„  +  f/)2  —  8rf#„        ,     . 

w  =  ■ — , (20) 

2d 


218  SERIES. 


RECURRING     SERIES. 

498.  A  Recurring  Series  is  one  in  which  each  term 
is  formed  by  multiplying  the  n  preceding  terms  each  by  a 
constant  multiplier,  and  adding  the  products.  These  multi- 
pliers are  called  the  Seale  of  the  Series, 

Thus,  in  the  series 

i,     4,     g,     16,     25, 

each  term  may  be  formed  by  multiplying  the  three  preceding  terms  by 
i,—3,  and  +  3,  respectively,  and  adding  the  products ;  as, 

1  x  1  t-  4(— 3)  +  9x3=  16, 

4x1  +  9  (—3)  +  16x3  =  25, 

9x1  +   16  (—3)  +  25  X3  =  36. 

499.  This  series  is  also  a  difference  series  of  the  second 
order,  and  any  term  or  the  sum  of  any  number  of  terms  may 
be  found  by  the  formulas  already  given. 

500.  There  are,  however,  cases  of  recurring  series  to 
which  the  method  of  differences  cannot  be  readily  applied. 
These  may  be  treated  by  finding  the  scale  of  the  series,  and 
using  it  to  find  the  term  or  sum  required. 

501.  To  find  the  scale  of  a  recurring  series,  assume  the 
series 

ah    a2,     (h>     (h,     ab,     afy    etc. 

If  each  term  depends  on  one  preceding  term  only,  we  have 
in  which  m  is  the  constant  multiplier.     This  gives 

On 

m  =  —  • 
<h 

502.  This  is  a  recurring  series  of  the  first  order,  and  is 
called  an  Equimultiple  Scries. 

It  is  also  called  a  Geometrical  Series  or  Progres- 
sion, and  is  the  most  important  of  recurring  series. 


SERIES.  219 

503.  If  each  term  of  the  series  depends  on  two  preceding 
terms,  the  series  is  of  the  second  order,  and  we  shall  have 

a3  =  mi«!  +  m2a2, 

and  «4  =  nixO^  +  m2o3, 

from  which  nix  and  mi}  the  scale  of  the  series,  are  found  to  be 

#2#4  —  o/  a2Oi  —  a,a4 

mi  =  -= ,  m2  =  —5 — 

af  —  axaz  a2l  —  aia3 

504.  In  the  same  way  the  scale  may  be  found  when  the 
series  is  of  a  higher  order.  If  the  order  of  the  series  be  not 
known,  we  may  assume  it  to  be  of  a  certain  order,  and  find 
the  scale. 

If  the  order  be  taken  too  high,  one  or  more  of  the 
multipliers  will  be  zero,  and  the  order  and  scale  will  be 
determined.  If  taken  too  low,  the  scale  found  will  be  incorrect, 
but  the  error  will  be  discovered  in  applying  the  multipliers. 

For  example,  to  find  the  scale  of 

i,     3,     6,     io,     15,     21,    etc., 
assume  the  series  to  be  of  the  second  order. 

We  shall  then  have,       mi  +  3m»  =    6, 
3?Wi  +  6m2  =  10. 

From  these  equations  we  find    m\  —  —  2, 

m2  =  |. 

In  attempting  to  extend  the  series  by  means  of  this  scale,  we  find  it 
fails ;  but  assuming  the  series  to  be  of  the  third  order, 

m-i  +  3?»2  +  6m3  =  10, 
3»?i  +  6m2  +  iom3  =  15, 
6m\  +  ioto2  +  15TO3  =  21. 

From  which,  m\  =  1,  mg  =  —  3,  m%  —  3,  which  is  the  trve  scale. 

The  same  would  have  been  found  if  we  had  assumed  the  series  to  be 
of  the  fourth  or  any  higher  order. 


220  SERIES. 

EXAMPLES. 

Find  the  scale  of  the  following  series: 

i.     Oi,    Oj  +  </,    <h  4-  2f/,    «!  +  2>d,    etc. 

-4ws.     —  I,    +2. 

«s  +  s«i  +  l°d,    etc-  ^W5-    r>  —  3)   +  3- 

3.  a3,      a3  +  a2,     a3  +  2^  +  a, ,      a3  +  3a2  4-  3a,  +  d, 

ai  +  4a-2  +  bax  +  4d,  az-\-sa.2+\oa},  +  iod,  a3-\-6a,+  15^  +  20^ 
fl3  +  7«2  +  21a,  +  35^  etc.     ^l«5.    —  1,    +4,   —  6,  +4. 

4.  a4 ,  a4  +  «3 ,  a4  +  2«3  +  a, ,  ai  4-  3«s  +  3«a  +  «i , 
<U  4-  4«s  +  6a2  4-  4«i  +  ^  «4  +  5*3  +  ioa«  4-  io«,  4-  5^> 
a4  4-  6a3  +  i5«s  +  2oai  +  15^  f/4  +  7«a  +  21a.,  4-35«i  +  35^ 
«4  +  8«3+28a.24-56rt,  -\-70d,     ai  +  gai-\-T>^aiJ<-^Aa\  +  12^d,    etc. 

-4res.    1,  —  5,   4-  10,  —  10,   4-  5. 

505.  The  student  will  observe  that  Examples  1  to  4  are 
general  expressions  for  difference  series  of  the  1st,  2d,  3d,  and 
4th  orders  respectively,  of  which 

d,    d,     d,     d,    etc., 

are  the  constant  differences.  These  constant  differences  form 
a  series  whose  scale  is  1,  and  we  may  call  it  a  difference  series 
of  the  zero  order. 

506.  From  these  solutions  we  infer  the  following  principles : 

i°.  Every  difference  series  is  also  a  recurring  series. 

20.  TJie  order  of  a  series  as  a  difference  series  is  one  less 
than  the  order  of  the  same  series  as  a  recurring  series. 

30.  All  difference  series  of  the  same  order  have  the  same 
scale  when  regarded  as  recurring  series. 

4°.  These  scales  are  the  same  as  the  coefficients  of  the  na 
poioer  of  a  —  x,  omitting  the  first,  changing  their  signs  and 
reversing  the  terms ;  n  being  the  order  of  the  series  considered 
as  a  recurring  series. 


SERIES, 


221 


507.  Any  recurring  series  not  having  such  a  scale  cannot 
be  a  true  difference  series ;  but  when  a  series  has  an  order  of 
differences  very  nearly  constant,  the  application  of  the  formulas 
for  difference  series  will  give  approximate  results. 

For  illustration  take  the  following  example: 


i.     Given     log.  200 
"     210 


2.30103, 
2.32222, 
220  =  2.34242, 
230  =  2.36173, 
240  =  2.38021, 
25°  =  2-39794, 


to  find  log.   205. 


Solution. — The  given  logarithms  form  a  series,  of  which  log.  205  is 
the  i\  term.     Finding  the  differences,  we  have, 

Series,  2.30103        2.32222         2.34242        2.36173         2.38021         2.39794 
1st  diff.,  .02119  .02020  .01931  .01848  .01773 

2d     "  —.00099       —.00089       —.00083       —.00075 

3d      "  .00010  .00006  00008 

4th    "  —.00004  .00002 

By  Formula  (A), 

1.1  1,1.3 

ai%  =  2.30103  +  ^(.02119)  +  *-*  (.00099)  +  -    —  (.00010) 
1-2  1-2-3 


+ 


—  (.00004). 


1.2-3.4 

/.  am  =  log.  205  =  2. 30103 +  .010595  +  .000124+  .000006  +  . 000002 
=  2.31175+, 

which  agrees  with  the  log.  205  from  the  table,  although  the  series  is  not 
a  perfect  difference  series. 

2.  Find  in  like  manner  log.  215. 

3.  Find  in  like  manner  log.  225. 

4.  Find  in  like  manner  log.  232. 


222  SEKIES. 

508.  The  Equimultiple  Series,  which  is  a  recurring 
series  of  the  first  order,  and  whose  terms  are  each  formed  by 
multiplying  the  preceding  term  by  a  constant  multiplier,  may 
be  written, 

«i ,    ajn,    axm2,    aAmz,     etc., 

in  which  «,  is  the  first  term  and  m  the  constant  multiplier. 
This  obviously  gives, 

an  —  axmn~x.  (D) 

Also,        Sn  =  eh  +-  axm  +  a,m2 -f  axmn~\ 

Multiplying  by  m, 

mSn  =  a-on  +  ajW2  +  a^n5  .  .  .  .  +  a1mn. 
Subtracting  the  first  from  the  second, 

mSn  —  S„  =  axmn  —  ax , 

alld  £  =  ^jrJh  =  •>&_■=£.  (E) 

m  —  i  m  —  i  ' 

509.  If  m  be  a  positive  integer,  greater  than  unity,  the 
series  will  be  increasing;  and  if  negative,  the  terms  will 
increase  numerically,  but  will  be  alternately  +  and  — . 

If  m  be  a  proper  fraction,  the  series  will  decrease ;  and  if 
the  number  of  terms  be  infinite,  an  will  be  by  Formula  (D), 

aM  =  (hm™  =  o, 

a,  (m™  —  i) 


and  S~  = 


m  —  i 


or  8n  =  -^— •  (E)' 

i  —  m 

510.  The  method  of  interpolation  already  given  applies 
to  all  series  when  a  formula  for  a„  can  be  found.  (Art.  496.) 
It  is  therefore  applicable  to  equimultiple  series,  and  we  may 
use  for  this  purpose  Formula  (D). 


SERIES.  223 

511.   To  find  a  term  between  the  4th  and  5  th,  make 


n  =  4f 

Then  ai}i  =  axmZYi    or    axwfl. 

To  interpolate  two  terms  between  the  4th  and  5th,  n  must 
be  made  4^  and  4!  successively. 

512.  Interpolation  may  also  be  performed  by  finding  the 
multiplier  of  the  series  formed  by  the  interpolated  and  adja- 
cent terms.  This  will  abbreviate  the  work  when  many  terms 
are  to  be  interpolated. 

Putting  m'  for  this  multiplier,  and  n'  for  the  number  of 
terms  to  be  interpolated  between  any  two  terms,  we  have 

1 
m!  =  ?w"7+I.  (F) 

Thus,  to  interpolate  3  terms  between  any  two  terms  of  the  series, 

1,     16,     256,    etc., 

i_  i 

in  which  m  —  16,  m'  =  i63+1  =  i6T  =  2. 

If  these  terms  be  inserted  between  16  and  256,  the  series  will  be, 

16,    32,     64,     128,     256. 


EXAMPLES. 

513.  Perform  the  examples,  p.  215,  by  the  principles  of 
recurring  series,  so  far  as  they  are  applicable;  also  the  follow- 
ing : 

1.  Find  «2o  and  S.i(t,  also  an  and  Sn  in 

1,     2,    4,     8,     16,    etc. 

2.  Find  aw  and  #10,  also  an  and  S„  in 

3,     9,     27,     81,     etc. 

3.  Find  am  and  S10,  also  an  and  S„  in 

5,     10,     20,     40,     etc. 


224  S  E  K  I  E  s . 

4.  Find  aa  and  8m  in 

8,    4,     2,     1,    etc. 

5.  Find  «„,  and  #„  in 

4,     1,     J,     etc. 

6.  Find  m!  for  interpolating  a  single  term  between  any 
two  of  the  last  series,  and  find  what  the  term  between  the  5th 
and  6th  will  be. 

7.  Interpolate  two  terms  between  each  two  of  the  series 
1,  8,  64,  etc.,  by  finding  m'. 

8.  Find  aw  and  Si0,  also  aB  and  S5  of  the  series 

h     —  3>     +9.     ~27,     +  etc. 

9.  Find  the  scale  of 

1,     2X,     3Z2,     5.T3,     io.r4,     21.T5,     43Z6, 
and  carry  the  series  to  the  10th  term. 

10.  Find  the  scale  and  an  of  the  series, 

2,     5.T,     8a;2,     1 1. r3,     142c*,     17a;5. 

11.  Find  from 

q^m*  —  1) 
* =       m-i     ' 

the  following  Formulas  for  Equimultiple  Series: 

4  =  ~  (1) 

a,  =  <yra  —  &(»i  —  1).  (2) 

(m  —  1 )  Sn  ,  x 

a,  (S„  -  a,)""1  =  «,.  (S„  -  a,,)"-'.  (4) 


S  EK1ES. 


225 


an  =  a-pn 


«i  +  Sn  (m  —  i) 
(»?  —  i)  Snmn~x 


ttn  mP  —  i 


$, 

= 

m  —  i 

& 

= 

anm  —  «! 

m  —  i 

& 

— 

an  (mn  — 

0 

7nn~1  (m  - 

-0 

& 

— : 

«„"_1  —  Oi 

T^i 

i 

i 

(5) 
(6) 

(7) 
(8) 

(9) 
(io) 

(») 

(12) 


«  "        —  «i 


m  = 


w 


8„  —  aK 
Sn  —  an 

= (r  • 


win m  =  i 

a,  a. 


mn  + 


a«  —  8, 


mn-l  — 


((„     —    Sn. 


log.  a„  —  loo-,  r?. 


log.  w 


w  = 


log.  an  —  log.  a, 


log.  (#„  -  ffl)  -  log.  (8n  -  a,) 


+  i. 


log.  1)1 

log.  a„  —  log.  [anm  —  (m  —  i)  S„] 
log.  w 


(i3) 

(i4) 

(i5) 
(16) 

(i7) 
(18) 


n  _  log-K  +  (w  -  i)^„]  -log.fli e  ( T g) 


+    I.       (20) 


226  SERIES. 


HARMONIC     SERIES. 

514.  An  Harmonic  Scries  is  one,  any  three  consecu- 
tive terms  of  which,  form  an  harmonic  proportion. 

515.  Three  terms  are  in   Harmonic  Proportion 

when  the  first  is  to  the  third  as  the  difference  of  the  first  and 
second  is  to  the  difference  of  the  second  and  third. 

Thus,  2,  3,  and  6  are  in  harmonic  proportion,  since 

2:6     =     (3  _  2)  :  (6  -  3). 

516.  Four  terms  are  in    Harmonic    Proportion 

when  the  first  is  to  the  fourth  as  the  difference  of  the  first 
and  second  is  to  the  difference  of  the  third  and  fourth. 

Thus,  3,  4,  6,  9,  are  in  harmonic  proportion,  since 
3:9     =     (4  -  3)  :  (9  -  6). 

517.  If  a,  b,  and  c  are  in  harmonic  proportion, 

a  :  c     =     (a  —  b)  :  (b  —  c), 
or  ab  —  ac  =  ac  —  be. 

Dividing  by  abc,  =•  =  =- That  is, 

J  c       b       b       a 

The  reciprocals  of  an  harmonic  series  form  an  equidifferent 
series ;  and,  conversely,  it  may  be  shown  that  the  reciprocals 
of  an  equidifferent  series  form  an  harmonic  series. 

518.  The  reciprocals  of  the  natural  numbers, 

T  I  I  I  I  I  pfp 

form  the  harmonic  series  to  which  the  name  was  first  applied, 
on  account  of  the  perfect  harmony  of  musical  strings  of  uniform 
size  and  tension,  whose  lengths  are  represented  by  the  terms 
of  this  series. 


SERIES.  227 

519.  To  find  the  nth  term  of  an  harmonic  series: 

Rule. — Find  the  nih  term  of  the  equidifferent  series  whose 
terms  are  the  reciprocals  of  the  terms  of  the  given  series  and 
take  its  reciprocal. 

Thus,  to  find  the  4th  term  of 

10,         12,         15, 

find  the  4th  term  of  ^,  ^,  TV,  which  is  £$,  and  take  its  reciprocal  20. 

The  series  is  then 

10,         12,         15,         20. 

520.  To  Interpolate  Harmonic  Means,  we  have 

the 

Rule. — Form  an  equidifferent  series  from  the  reciprocals 
of  the  terms  of  the  harmonic  series,  and  interpolate  the  corre- 
sponding equidifferent  means  and  take  their  reciprocals. 

Thus,  to  interpolate  two  harmonic  means  between  10  and  20,  inter- 
polate two  equidifferent  means  between  -^  and  ^V  an<i  ta^e  the 
reciprocals. 

EXAMPLES. 

1.  Find  the  5th  term  of  the  harmonic  series,  \,  \,  £,  etc. 

2.  Insert  two  harmonic  means  between  \  and  ^. 

3.  Insert  3  harmonic  means  between  5  and  25. 

4.  Show  that  the  equimultiple  mean  between  two  quan- 
tities is  an  equimultiple  mean  between  their  equidifferent  and 
harmonic  means. 

DEVELOPMENT     OF     FORMULAS. 

521.  Develop  from  the  solution  of  the  following  problems 
the  formulas  by  which  all  similar  problems  may  be  solved. 

1.  What  is  the  amount  (a)  of  a  note  for  p  dollars,  at 
compound  interest  t  years,  at  r  per  cent? 

2.  What  payment  (p)  made  annually  for  t  years  will  pay 
the  sum  s  dollars,  with  compound  interest  at  r  per  cent? 

3.  What  sum  (s)  put  at  compound  interest  at  r%  will 
amount  to  a  in  t  years  ? 


228  SERIES. 

4.  What  is  the  present  worth  (w)  of  a  sum  (s)  due  in 
t  years,  money  being  worth  r%  compound  interest? 

5.  What  is  the  present  worth  of  a  perpetual  annuity  («), 
to  commence  in  t  years,  at  r%  compound  interest  ? 

6.  What  is  the  present  worth  of  an  annuity  (a),  to 
commence  in  t  years,  and  to  continue  t'  years,  allowing  r% 
compound  interest  ? 

7.  Find  the  amount  of  a  note  for  £500  on  interest  3  years, 
at  compound  interest  at  6%. 

8.  Find  the  present  worth  of  $1000  due  in  t>\  years,  allow- 
ing compound  interest  at  4%. 

9.  Find  the  sum  that  will  amount  to  $1000  in  5  years,  on 
interest  at  $%. 

10.  Find  the  present  worth  of  an  annuity  to  commence  in 
2  years  and  to  continue  10  years. 

11.  Find  the  present  worth  of  an  annuity  in  arrears  3  years 
and  to  continue  7  years  longer. 

Note. — Use  formula  from  Problem  6. 

12.  Find  the  present  worth  of  a  perpetual  annuity  of  8100 
to  commence  in  5  years,  at  6%  compound  interest. 

13.  The  same  as  above,  to  commence  now. 

14.  The  same  as  above,  to  commence  5  years  ago. 

15.  Find  the  time  when  an  annual  payment  of  $100  should 
have  begun  to  cancel  at  maturity  a  note  for  $500  given 
Jan.  1,  1875,  and  due  Jan.  1,  18S0,  with  compound  interest 
at  $%. 

16.  A  man  travels  from  a  certain  point  northward  10  miles 
the  first  day,  9  miles  the  second,  8  miles  the  third,  and  so  on, 
continuing  the  series  by  the  same  law.  How  far  north  will  he 
be  at  the  end  of  5  days?  How  far  at  the  end  of  10  days  ? 
11  days?  20  days?  22  days?  30  days?  How  far  north 
will  he  travel  the  15th  day? 

17.  A  man  travels  as  above  10  miles  the  first  day.  and  each 
succeeding  day  -^  as  far  as  the  day  before.  How  far  north 
will  he  be  in  5  days?  In  10  days?  How  long  would  it 
require  for  him  to  travel  25  miles?  How  far  would  he  travel 
if  he  should  continue  forever? 


SERIES.  229 

18.  A  ball  falling  from  the  height  of  ioo  feet  rebounds 
50  feet.  If  it  continue  to  rebound  one  half  the  distance  it 
falls,  how  many  times  will  it  rebound  ?  How  far  will  it  move 
before  coming  to  rest  ? 

522.    The  following  identical  equation 

i 


m(m+p)(m  +  2p)  .  .  {m  +  rp)     rp  \  m{m+p)  .  .  [m  +  (r—i)p~\ 


« I 

(m+p)  (m  +  2p)  .  .  (m  +  rp)  \ 


(E) 


furnishes  a  method  for  the  summation  of  series  whose  terms 
are  of  the  form  of  the  first  member. 

The  following  examples  will  illustrate : 

1.  Find    8n   and    Sm    of  —  +  -*-  +  —  +  etc. 

1.2       2-3       3-4 

Solution. — Here 

a  =  1  ;         m  =  1,  2,  3,  etc.  ;        p  =  1  ;         and     r  =  I. 

Substituting  in  2d  member  of  (E), 


S, 


'n  =  \(\  +  l  +  i  .  .  .   +  -  -i-i  ...   - — ) 

1 V  n      -  n      11  +  1/ 


—  1 —  = .    Ans. 

n+i       n + 1 


If   n  =  co  ,    1 =  1 =  I  : 

n+l  co 


2.  Find   S„    of   -i-g  +  ^—  +  — --7  +  etc. 
3-8       6-i2        9-16 

Solution. — Multiplying  the  series  by  12  gives 

111 

-  + H +  etc., 

1-2       2-3       3-4 

which  is  the  same  as  in  Ex.  1. 

£L  =  A. 


230  SERIES. 

3.  Find   S„    of ^—  +  --- -5—  4-  — -_  +  etc 

1-2-3       2-3-4       3-4-5 

Solution,     a  —  4,  5.  6,  etc. ;    to  =  1,  2,  3,  etc.  ;    p  =  1  ;    r  =  2. 
Substituting  in  (E), 

5.  =  I  (-*-  +  -5-  +  -^  +  -*-  +  etc.  -  -4-  -  -A.  -  -5_  _  etc.) 
\i  -2       2.3       3-4       45  23       3.4       4.5  / 

=  A  (— -  +  —  4-  —  +  —  +  etc.  ). 
"Vi  -2       2-3       3-4       4-5  / 

Applying  (E)  again  to  this  series,  beginning  with  the  term , 

we  have, 

a  —  1  ;        to  =  2,  3,  4,  etc.  ;        p  =  1 ;        r  =  1. 

$»  =  i  +  I  +t  +  i  +  etc-  -  s  -  1  -  5  -  etc  =  f 


5 


=  h  (—  +  -)  =  f-    -4«»- 

-  \i  •  2      2/       4 


Find   #,„  of  the  following: 

1  4  7 

4.        H 1 £ —  +  etc. 

i-3-5       3-5-7       5-7-9 

5.  -i ^-+^ 5_-  +  ete. 

3-5       5-7        7-9       9  '  IJ 

1             1  1 

6. 1 h  --    -  4-  etc. 

i-3       -'-4       3-5 

1  1  1 

7. Y etc. 

i-3       2-4       3-5 

8.     -1-  4-  -±-  +  — i--  4-  etc. 
i-5       5-9       9-^3 

1  1  1 

+  -  +  -  -2  +  etc. 


1  •  2  •  3  •  4       2-3.4.5        3-4-5"6 

1  1  1 

io.       l.  _ 1 4-  etc. 

1  • 3  • 5  *  7        3-5-7-9       5  •  7  •  9  "  T  T 

I 2  22  32 

11.        —  +  -  H 2 -,  4-  etc. 

1.2.3.4       2-3.4.5       3 -4.  5  •  6 

I2  22  32 

12. y-  -  H —       +  etc. 

1-3-5-7       3-5-7-9       5  -7  -9  •  " 


SERIES. 


231 


REVERSION     OF     SERIES. 

523.  When  we  have  y  =  a  series  which  is  a  function  of 
x,  finding  x  as  a  function  of  y  is  called  Reverting  the 
Series. 


Given,      y  =  x  +  x2  +  x3  +  a;4  +  x5  -f  etc., 

to  revert  the  series. 
Solution. — Assume 

x  =  Ay  +  By2  +  Cy3  +  Dy*  +  etc. 
Substituting  this  value  of  a;  in  (i), 


y  =  Ay  +  B 

+  A2 


if  +  C 
+  2AB 

+  ^l3 


y3  -\-  D       \yi  +  etc. 

+  B2 
+  3^25 
+  J4 


Hence,  Art.  419,  A  =  1 , 

5  +  .42  =  o;  .-.  B  =  -1. 

C  +  2.-l£  +  ^l3  =  o;  .-.  C  =  1. 

D  +  2AC+B2  +  sA2B  +  Ai  =  o;  .-.  #  =  —  1. 

The  law  of  the  series  is  evident  and  we  may  write 

x  =  y  —  if  +  y3  —  y*  +  f  — etc- 


(0 


(*) 


EXAMPLES. 

Revert  the  following  series : 

1.  y  =  x  +  2X2  +  3.r3  -f  4.T4  +  etc. 

2.  y  =  1  —  x  +  x2  —  x3  +  .r4  —  eti 

3.  ?/  =  x  —  \x2  +  Ice8  —  \x^  +  etc. 

4.  y  ■=  1  +  x  +  2x2  +  $x3  +  etc. 


CHAPTER    XX. 

LOCI     OF     EQUATIONS. 


525.  We  have  seen  that  an  equation  with  two  or  more 
unknown  quantities  is  indeterminate ;  that  is,  there  are  no 
definite  values  that  can  be  assigned  as  the  only  values  of  these 
quantities.     (Art.  241.) 

For  example,  in  the  equation  x  +  y  =  5,  we  may  have 
x  =  1  and  y  =  4,  x  =  2  and  y  =  3,  x  =  3  and  y  =  2, 
and  so  on  indefinitely. 

And  x  not  only  may  be  any  integral  number  between 
+  00  and  —  co,  but  it  may  have  any  value,  integral,  frac- 
tional, or  incommensurable  between  those  limits.  Hence,  x 
and  y  are  variables,  and  pass  from  one  value  to  another  by 
infinitesimal  increments  ;  that  is,  they  pass  through  all  inter- 
mediate values,  as  a  point  moving  from  one  position  to 
another  along  a  line  passes  through  all  intermediate  points. 
As  the  number  of  these  points  is  infinite,  so  the  number  of 
different  values  of  a  variable  is  infinite.     (Art.  390.) 

526.  The  relation  of  an  equation  containing  two  variables, 

to  a  line  considered  as    the    path    of   a 
/         point    moving    in    a    plane    surface,    is 
more  fully  illustrated   by  the  following 
method. 


e/P 
d/ 


Fig.  I. 


1$ 


Q 


527.   Assume  the  two  lines  XX'  and 
YY'  (Fig.    1)    at    right    angles    to    each 
other,    intersecting    at    A.        Take    any 
equation  with  two  variables,  as 
y  =  2X  +  3. 


LOCI     OF     EQUATIONS.  233 


In  this  eqn< 

ition, 

If    x  =  o, 

y  =  3- 

If 

X  =    —  I, 

y  = :   I. 

If      X  =    I, 

y  =  5- 

If 

X    =    2, 

?/  =■  —I 

If      X  =   2, 

y  =  7- 

If 

£C  =  —3, 

y  =  — 3 

If    *  =  3, 

y  =  9- 

If 

as  =  —  4j 

2/  =  -5 

If    x  =  4, 

y  =  ii- 

If 

x  =  —  5, 

y  =  -7 

If    x  =  5, 

!/  =  J3- 

If 

x  =  — 6, 

y  =  —9 

If,  now,  we  adopt  some  convenient  unit,  and  measure  the 
positive  values  of  x  from  A  towards  X  (the  negative  values 
being  of  course  measured  in  the  opposite  direction),  we  shall 
find  on  XX'  the  several  points  i,  2,  3,  4,  etc.,  —  1,  —2, 
—3,  —4,  etc. 

If  from  each  of  these  points  we  erect  a  perpendicular  equal 
to  the  corresponding  value  of  y,  we  shall  have  the  lines  bi, 
C2,  d3,  and  b'  (  —  1),  c'  ( —  2),  d'  (  —  3),  etc. 

If  we  have  carefully  drawn  the  lines  to  the  proper  measure, 
we  may  now  place  a  ruler  upon  them,  and  connect  all  the 
points  a,  b,  c,  d,  etc.,  by  one  straight  line  PQ. 

We  may  also  take  fractional  values  for  x  and  find  correspond- 
ing values  for  y  from  the  given  equation,  and  all  the  lines 
representing  the  values  of  y  will  terminate  upon  this  line  PQ. 

In  like  manner,  any  point  on  the  line  PQ  represents  a  set 
of  values  for  x  and  y  by  its  distances  from  the  lines  YY' 
and  XX'. 

Hence,  the  equation  y  =  2x  +  3  is  said  to  be  the 
equation  of  the  line  PQ. 

528.  It  is  important  for  the  student  to  become  familiar 
with  the  definitions  of  the  following  terms  used  in  discussions 
of  this  kind. 

Def.  1.   The  Axes,  or  Axes  of  Reference  are  the 

assumed  lines  XX'  and  YY'. 

2.  Tile  Or  iff  in  is  their  point  of  intersection  A. 

3.  The  Axis  of  Abscissas  is  the  line  XX'. 


234  LOCI     OF     EQUATION'S. 

4-  TJie  Axis  of  Ordinates  is  the  line  YY'. 

5.  The  Ordinate  of  a  point  is  its  distance  from  the 
axis  of  abscissas.  Thus  the  ordinates  of  b,  c,  d,  etc.,  Fig.  I, 
are  bi,  C2,  d3,  etc. 

6.  The  Foot  of  the  Ordinate  is  the  point  where  it 
meets  the  axis  of  Abscissas. 

7.  Tlte  Abscissa  of  a  point  is  the  distance  from  the 
origin  to  the  foot  of  its  ordinate,  or  the  distance  of  the  point 
from  the  axis  of  ordinates. 

8.  The  Co-ordinates  of  a  point  are  its  abscissa  and 
ordinate. 

9.  TJie  Locus  of  an  Equation  is  the  line  which  the 
equation  represents. 

10.  Constructing  a  Locus  is  drawing  the  line  (as 
shown  above)  represented  by  an  equation. 

11.  Abscissas  are  positive  when  measured  to  the  right, 
and  negative  when  measured  to  the  left  of  the  axis  of 
ordinates. 

12.  Ordinates  are  positive  above  and  negative  below  the 
axis  of  abscissas. 

529.  The  four  parts  into  which  the  plane  is  divided  by 
the  axes  are  called  the  first,  second,  third,  and  fourth  angle 
respectively,  beginning  with  the  angle  on  the  right  of  the 
axis  of  ordinates  and  above  the  axis  of  abscissas,  and  going 
round  to  the  left. 

Thus,  YAX  is  the  first,  YAX'  the  second,  VAX'  the  third,  and  VAX  the 
fourth  angle. 

530.  Abscissas  are  usually  represented  by  the  letter  x, 
ordinates  by  g. 

In  the  first  angle  x  and   g   are  both  positive. 
In  the  second  angle  x  is  negative  and  g  positive. 
In  the  third  angle  x  and   g  are  both  negative. 
In  the  fourth  angle  x  is  positive  and  g  is  negative. 


LOCI     OF     EQUATIONS.  235 

531.   Construct  the  loci  of  the  following  equations : 

2.  y  —  2>x  +  2.  5.     y  =  —  3x  +  2. 

3.  y  =  2X  —  I.  •  6.     y  =  —  x. 

4.  V  =  3»-  7-     2/  =  2. 

Note. — The  last  gives  a  line  every  point  of  which  has  its  ordinate  2. 

Equations  of  the  first  degree  always  represent  straight  lines.  They 
need  not  be  in  the  form  given  above,  but  may  for  convenience  be  put  in 
that  form  before  constructing  the  loci. 

Construct  the  following  loci : 
8#     x-y  _x  +  y_ 


2                 6 

9- 

2X  —  3?/   =   3 

10. 

X 

5  =0. 

2y 

532.  When  we  make  x  =  o,  we  find  the  point  where  the 
line  crosses  the  axis  of  ordinates,  aud  when  y  .=  o,  the  point 
where  it  crosses  the  axis  of  abscissas. 


y  =  2x  +  3, 


Thus  in  the  equation 
if  y  =  o,   we  have, 

2X   +    3    =    O, 

and  x  =  —  f. 

This  is  the  distance  A  (—  f)  (Fig.  I),  and  it  is  the  root  of  the  equation 
2.v  +  3  —  0.  Thus  we  see  how  we  may  construct  the  roots  of  equations 
having  one  unknown  quantity. 

11.  Construct  the  root  of 


3 


SOLUTION. 

Put  the  equation  in  the  form, 

x  —  4      x  —  2       x 


236 

X  —  4      x  —  2       X  —  8 

Make  y  = ; — , 

2  3  6     * 

and  construct  the  locus.     The  abscissa  of  the  point  where  this  locus  cuts 
the  axis  of  the  abscissas  will  be  the  root  required. 


236 


LOCI     OF     EQUATIONS 


533.  It  appears  from  this,  that  equations  of  the  first 
degree  should  give  straight  lines;  for  if  the  line  could  cut 
the  axis  of  abscissas  a  second  time,  a  second  root  would  be 
found,  which  is  not  possible  for  such  an  equation.    (Art.  232.) 


12.  Construct  the  locus 

y  =  xz  —  2X2 
and  find  the  real  roots  of 


x  +  2, 


xz  —  2a;2  —  x  +  2  =  o. 

Note  — It  is  to  be  observed  that  the  imaginary  roots  cannot  be  thus 
constructed. 


Fig.  II. 


(\ 


I    2 


Fig.  Ill 


Fig.  IV 


Assuming  the  axes   XX'   and  YY'   (Fig.  II),  and  making 


X    = 

0, 

y 

= 

2 ; 

X    = 

I 

2  J 

y 

= 

9 . 

X    = 

I, 

y 

= 

0 ; 

X    = 

—I, 

y 

= 

0 ; 

X    = 

lh 

y 

= 

5  • 
— s  > 

X    = 

-I* 

y 

= 

— 4|; 

X    — 

2, 

y 

= 

°; 

X    = 

—  2, 

y 

= 

—12; 

X    = 

3' 

y 

= 

8; 

X    = 

4, 

y 

= 

3°- 

LOCI     OF     EQUATIONS, 


237 


The  roots  required  are  2,  1,  and  —  1,  represented  by  the 
distances  (A2),  (Ai),  and  [A  (—  1)]. 

This  example  illustrates  the  three  roots  of  the  equation  of 
the  third  degree,  showing  how  the  line  is  cut  by  the  axes  of 
abscissas  three  times. 

13.  Construct  the  locus 

xs  +  3Xz  +  6x  +  1  =  y, 
and  find  the  value  of  x  when   y  =  o.     (See  Fig.  III.) 


Fig.  VI 


Fig.  VII. 


UJ 


lY 


Making 


X  =  0, 

y  =  1; 

X  =  1, 

y  =   n; 

X    =     — I, 

2/  =   -3; 

X    =    2, 

^  =  33  5 

X    =     — 2, 

#  =   —7; 

x  =   —3, 

2/  =   ~l7 

By  constructing  these  points  and  others  intermediate,  by 
making  x  =  \,  —  \,  etc.,  and  sketching  in  the  curve  to 
join  the  points,  we  may  measure  the  distance  AB,  which  will 
give  us  approximately  the  real  root  of  the  equation. 

The  construction  also  shows  the  other  roots  to  be 
imaginary. 


238  LOCI     OF     EQUATIONS. 

In  like  manner  construct  and  find  real  roots  of 
y  =  x 


14- 
IS- 
16. 

17- 
18. 
19. 


r2  -  5.     (See  Fig.  IV.) 
y  =  a?  —  2x*  +  1.     (See  Fig.  V.) 
y  —  &  _  q&  _  3x  +  18.     (See  Fig.  VI.) 
y  —  tf  —  2x*  +  i.     (See  Fig.  VII.) 
y  =  x*  -  2X*  —  1.     (See  Fig.  VIII.) 
y  —  X4  _  2X2  +  2.     (See  Fig.  IX.) 


20.     y  =  x5 — 3a4 — 5»3+  i5z2  +  4£— 12.     (See  Fig.  X.) 


Fig.  VIII. 


-x- 


UJ 


Fig.  IX 


Fig.  X 


Note. — The  object  of  this  chapter  is  to  furnish  illustrations  of  the 
principles  to  be  developed  relating  to  the  general  Theory  of  Equations, 
and  not  to  give  a  practical  method  of  finding  the  roots  of  equations. 


CHAPTER    XXI. 

THEORY     OF     EQUATIONS. 

534.  When  a  problem  involving  but  one  unknown  quantity 
produces  an  equation  of  the  first  or  second  degree,  we  can 
easily  solve  it  in  a  general  manner,  by  using  general  symbols, 
as  letters,  for  the  known  quantities,  thus  obtaining  a  formula 
by  which  the  unknown  quantity  may  be  found  for  all  special 
cases  of  the  problem,  by  mere  arithmetical  computation.  Such 
formulas  were  obtained  in  Arts.  240,  340. 

535.  But  when  a  problem  gives  rise  to  an  equation  of  the 
third  or  fourth  degree,  the  same  may  be  done,  but  with  much 
more  difficulty,  except  in  special  cases.  Indeed,  so  complicated 
is  the  reduction  of  the  general  equation  of  the  fourth  degree, 
that  it  is  seldom  employed  in  practice. 

536.  For  equations  above  the  fourth  degree,  no  general 
method  of  reduction  has  yet  been  found.  But  when  such 
equations  arise,  by  putting  for  the  known  quantities  numerical 
values,  instead  of  general  symbols,  thus  forming  equations  with 
numerical  coefficients,  called  numerical  equations,  the  real 
roots  may  be  readily  found. 

By  this  method,  the  arithmetical  part  of  the  solution  of  the 
problem  is  performed  before  the  algebraic,  and  the  equation 
must  be  reduced  for  each  special  case  of  the  problem. 

537.  The  object  of  the  present  chapter  is  the  discussion 
of  the  methods  employed  to  find  the  real  roots  of  numerical 
equations  of  a  higher  degree  than  can  be  readily  reduced  by 
the  methods  already  given.  Observe  we  do  not  say  methods  of 
reducing  higher  equations,  for  the  reduction  of  an  equation 


240  NUMERICAL     HIGHER     EQUATIONS. 

implies  the  use  of  Axiom  i,  Art.  38,  in  such  manner  as  to 
bring  out  the  unknown  quantity  as  an  explicit  function  of  the 
known  quantities.  This  is  not  done  with  higher  equations; 
but  by  various  devices  depending  on  the  General  Theory 
of  Equations;,  the  real  roots  are  discovered  without  a 
process  of  reduction. 

538.  To  facilitate  the  discovery  of  the  real  roots  of 
Higher  Numerical  Equations,  they  are  reduced  to 
the  form 

xn  _j_  A&n-1  +  AaJK~% in_,x  +  Am  =  o.        (1) 

in  which  Ai,  Ai}  etc.,  and  n  are  integral  and  the  exponents 
are  all  posit  ire. 

The  reductions  necessary  to  give  this  form  to  an  equation 
may  be  any  or  all  of  the  following: 

1.  To  make  the  exponents  positive. 

2.  To  make  the  exponents  integral. 

3.  To  make  the  coefficient  of  xn  unity. 

4.  To  make  the  coefficients  Ax,  Ai}  etc.,  integral. 

These  transformations  may  be  made  as  follows  : 

539.  To  Make  the  Exponents  Positive. 

Eule. — Multiply  the  equation  byxvrith  a  positive  exponent 
equal   numerically   to   the   largest    negative    exponent   in   the 

equation. 

Note. — This  will  evidently  accomplish  the  desired  transformation, 
and  will  not  affect  the  roots  of  the  equation,  since  both  members  are 
equally  affected.     It  is  in  fact  only  clearing  the  equation  of  fractious. 

540.  To  Make  the  Exponents  Integral. 

Rule. — Multiply  each  exponent  by  the  least  amnion  multi- 
ple of  the  denominators  of  the  fractional  exponents. 

This  will  obviously  make  the  exponents  integral,  but  it  will 
at  the  same  time  change  the  roots  of  the  equation.     It  will 


NUMERICAL     HIGHER     EQUATION'S.  241 

therefore  be  of  no  advantage  to  find,  the  roots  of  the  trans- 
formed equation,  unless  we  can  ascertain  what  function  its 
roots  will  be  of  the  primitive  roots. 

This  we  may  easily  do,  for  if  m  be  the  common  multiple 
used,  the  transformation  will  be  made  by  substituting  ym 
for  x.     Thus, 

Let  the  original  equation  be 

ill 

xa  +  Axxh  -f  A,xc  +  Az  =  o. 

Substituting  ym  =  x, 

III 
{ymf  +  Al{y™)h  +  Ai(tf»y+  A3  =  o. 

Eeducing, 

m  m  m 

if  +  A^xf  +  A*y~e  +  A,  =  o. 

If  m  be  made  a  multiple  of  a,  b,  and  c,  that  is,  abc,  the 
equation  becomes 

ybc  +  A^y™  +  A,yab  +  A3  =  o, 

in  which  each  exponent  has  been  multiplied  by  m  =  abc. 

But  since  y™  =  x,  the  values  of  y  must  be  raised  to  the 
mth  power  to  give  the  values  of  x.     Hence, 

When  the  exponents  of  x  in  an  equation  are  multiplied  by 
m,  the  roots  of  the  transformed  equation  raised  to  the  mth 
power  will  be  the  roots  of  the  primitive  equation. 

541.    To  Make  the  Coefficient  of  ,xn  Unity. 

Rule. — Divide  the  equation  by  that  coefficient. 

This  evidently  does  not  affect  the  roots. 
11 


242  NUMERICAL     11  I  <  MI  E  R     E  Q  U  A  T I O  N  S . 

542.  To  Make  the  Coefficients  integral  without  Changing  the 
Coefficient  of  ae71. 

Utile.  — Multiply  the  coefficient  of  xn~x  by  kj  that  of  xn~2 
by  J&;  that  of  aP~3  by  k\  and  so  on,  to  the  coefficient  of  .'". 
making  k  such  a  number  as  will  render  each  coefficient 
integral. 

v 
This  rule  is  obtained  by  substituting  |  for  x,  and  clearing 

of  fractions ;  thus, 

V 
Substituting  t  =  «  in  (0> 

yn  yn-1  yn-2 

f-    +    AA :    +  AA ; J    .    .    .     .      4-    An    =    O. 

Clearing  of  fractions, 

if  +  A,kyn^x  +  AJcttf1-*  .  .  .  .    +  AJn  —  o. 

If,  now,  any  of  the  coefficients,  Ax,  A,,  etc.,  are  fractional, 
a  value  for  k  maybe  taken  such  as  to  remove  the  denominators. 
Since 

V  7 

I  =  x,  y  =  kx. 

That  is,  the  roots  have  been  multiplied  by  k.  and  when 
found  must  be  divided  by  k.  to  give  the  primitive  roots. 

EXAMPLES. 
Reduce  the  following  equations  to  the  form  (i), 

B  4  l 

I.     2X*  —  $x*  +  7£3  -|  =  o. 

Solution. — Multiplying  the  exponents  of  x  bv  3,  we  have, 

2y5  -  $if  +  Hi  -  \  =  o, 
in  which  yz  =  X.     (Art.  540.)     Dividing  by  2  gives, 

y-  -  f*/4  +  ly  ~  I  =  o.     (Art,  541.) 


NUMERICAL     HIGHER     EQUATIONS.  243 

Applying  Rule,  Art.  542, 

,5       5*  4   ,    7**.       3*5 

s-1 g*  +  —  2  —  — —  =  O, 

228 

in  which  z  =  ley.    Making  k  =  2,  to  cancel  denominators, 

z5  —  5s4  +  56s  —  12  =  0. 

in  which  z  =  2y,  and  since  y3  =  x, 

z  =  2xK 

2.     32^  —  5$*  -j-  8a;  —  2$  —  1  =0. 

3-     5*r*  —  3«*  +  9^*  +  5  =  o. 
4.     |aj*  —  iJ  +  |o£  —  \  =  o. 

543.  We  have  thus  seen  how  any  equation  with  a  single 
unknown  quantity  having  rational  coefficients  and  exponents, 
may  be  reduced  to  the  form  (1)  (in  which  the  coefficient  of  xn 
is  unity,  the  other  coefficients  are  integral,  and  the  exponents 
integral  and  positive),  without  making  any  unknown  change  in 
the  roots. 

We  now  proceed  to  discuss  this  equation,  for  the  purpose 
of  discovering  its  real  roots.  For  the  sake  of  brevity  we  shall 
use  for  it  the  symbol 

f(x)  =  o. 


I.     DIVISIBILITY. 
Theorem  I. 

544.  If  f(x)  be  divided  by  x  —  a,  the  remainder  will  be 
f{a),  that  is,  it  will  be  what  f{x)  becomes  when  a  is  substituted 
for  x. 

Demonstration.— Let  q  be  the  quotient  and  r  the  remainder 
obtained  by  the  division.     Then 

f(x)  =  (x  —  a)q  +  r, 

an  equation  true  independently  of  the  value  of  x.     It  is  therefore  true 
when  x  —  a.     Substituting  a  for  x,  we  have, 

/(«)  =  r. 


244  NUMERICAL     HIGHER     EQUATIONS. 

Cor.  i.  —  When  a  is  a  root  of  f(x)  =  o,  fix)  is  divisible 
by  x  —  a,  and  not  otherwise. 

For  if  a  be  a  root,  f(a)  =  o.  (Art.  217.)  .*.  r  =  o,  and  the  division 
is  complete.  But  if  a  be  not  a  root,  f(a)  is  not  o,and  r  is  not  o.  and  the 
division  is  not  perfect. 

Cor.  2. — If  a  be  a  root  of  f(x)  =  o,  a  is  a  factor  of  the 
absolute  term. 

For,  it  is  evident  that  if  f(x)  is  divisible  by  x  —  a,  the  absolute  term 
is  divisible  by  a. 

545.  This  theorem  gives  an  easy  method  of  determining 
whether  any  number  be  a  root  of  an  equation. 

Try  what  factors  of  the  absolute  terms  are  roots  of  the 
following  equations : 

1.  x3  —  &x2  +  n.c  +  20  =  o. 

2.  x5  —  x4  —  252^  +  85a;2  —  962;  +  36  =  o. 

3.  .t5  —  1  =  o. 

4.  a4  —  2X5  +  8x  —  16  =  0. 

5.  x3  —  iiz2  +  433  —  65  =  o. 

6.  x5  —  ix3  -\-  2X  -\-  15  =  o. 

In  making  these  trials,  use  the  method  of  synthetic  division. 
(Art.  151.)     Thus,  to  try  —3  in  Ex.  6,  we  write, 

1+0  —  7  +  0+     2  +  15  I  —  3 

—  3  +  9  —  6+18  —  60 

—  3  +  2  —  6  +  20  —  45.  remainder. 
The  remainder  is  —45.  and  —3  is  not  a  root. 

546.  "We  have  also  from  this  theorem  a  method  of  deter- 
mining the  value  of  a  function  when  any  number  is  substituted 
for  x.  Thus,  when  —  3  is  substituted  for  x  in  Ex.  6,  the 
first  member  reduces  to  —  45. 

Note. — It  should  be  observed  that  this  theorem  is  applicable  to  all 
equations  and  not  merely  to  those  reduced  to  the  form  of  (1). 

Find  the  value  of  the  first  member  of  each  of  the  preceding 
equations,  when  1,  2,  3,  —  1,  —  2,  and  —  3  are  substituted 
for  x. 


NUMERICAL     HIGHER     EQUATIONS.  245 

II.      NUMBER     OF     ROOTS. 

Theorem  II. 

547.  Every  equation  of  an  integral  degree  has  the  same 
number  of  roots  as  there  are  units  in  its  degree. 

Demonstration. — To  prove  this  theorem,  it  is  necessary  to  assume 
that  every  such  equation  has  at  least  one  root,  since  the  proof  of  this  is 
not  within  the  scope  of  elementary  Algebra.  But  with  this  assumption 
we  may  easily  prove  the  theorem,  for,  representing  this  root  by  ctu  and 
dividing  the  first  member,  when  put  in  the  form  f(x)  =  o,  by  x  —  ax  , 
we  have 

(x—afiq  =f(x)  =  o, 

which  is  satisfied  when  x  —  a\  =  o,  or  when  q  =  o.  But  q  is  a  function 
of  x  one  unit  lower  in  degree  than  the  original  function,  and  q  —  o  has 
by  the  assumption  one  root,  which  may  be  divided  out  as  before,  and 
this  process  may  be  continued  so  long  as  q  is  a  function  of  x  ;  hence  the 
theorem  is  true,  and  f(x)  —  o  has  n  roots  and  no  more. 

Cor.  i. — The  first  member  off(x)  =  o  is  the  product  of 
the  n  binomial  factors, 

(x  —  «j)  (x  —  a.2)  (x  —  a3)  .  .  .  .  (x  —  an), 

in  which  a1}  «2,  etc.,  are  its  roots. 

Note. — It  is  not  to  be  understood  that  n\ ,  a?,  etc.,  are  all  different 
numbers,  for  there  is  nothing  in  the  proof  of  the  theorem  to  forbid  their 
being  all  equal. 

III.     FORMATION     OF     EQUATIONS. 

Cor.  2. — An  equation  mag  be  formed,  having  any  roots 
whatever,  by  subtracting  each  root  from  x,  and  putting  the 
product  of  the  binomials  thus  formed  equal  to  zero.     (Cor.  i.) 

Cor.  3. —  The  coefficients  of  the  several  powers  of  x  in  the 
first  member  of  f(x)  =  o,  will  be  as  follows : 

Of  xn~x,  the  sum  of  the  roots  with  their  signs  changed. 

Of  xn~2,  the  sum  of  the  products  of  the  roots  taken  two  and 
two.     (Art.  z&T,.) 


246  NUMERICAL     HIGHER     EQUATIONS. 

Of  xn~3,  the  sum  of  the  products  of  the  roots,  with  their 
sign*  changed,  taken  three  and  three ;  and  in  general, 

The  coefficient  of  x"  i>  will  be  the  sum  of  the  products  of  the 
roots,  with  /heir  signs  changed,  taken  p  at  a  time. 

This  follows  from  the  binomial  formula,  in  which  it  is  evident  that 
the  coefficients  are  formed  in  the  same  manner  as  in  /  ■')•  Observe 
that  in  forming  f(x)  by  multiplying  n  binomial  factors, the  second  terms 
of  these  factors  are  the  roots  with  their  signs  cfiangi  d ;  hence  the  products 
which  form  the  several  coefficients  are  all  formed  by  the  mots  with  their 
signs  changed,  but  those  products  which  have  an  <  ven  number  of  factors, 
as  those  taken,  2,  4,  6,  etc ,  at  a  time,  will  have  the  same  sign  whether  the 
signs  of  the  roots  be  changed  or  not. 

By  this  corollary  an  equation  having  any  given  roots  may  be  formed 
by  changing  the  signs  of  the  roots,  and  forming  the  several  coefficients 
in  accordance  with  this  law. 

Cor.  4.  The  absolute  term  or  coefficient  of  x°  is  the  product 
of  all  the  roots  ivith  their  signs  changed. 

Note— Let  the  student  notice  how  these  principles  applied  to 
equations  of  the  second  degree  give  the  same  results  as  those  already 
obtained  in  Chapter  XI,  Art.  329. 

548.  The  truth  of  Theorem  II  is  illustrated  by  the  loci 
of  equations,  where  the  number  of  roots  is  represented  by  the 
number  of  times  a  straight  line  can  cut  the  locus  of  the  equa- 
tion. This  will  be  seen,  by  an  examination  of  the  loci  already 
constructed,  to  be  equal  to  the  number  of  units  in  the  degree. 

Observe  that  it  is  not  the  number  of  times  the  axis  of 
abscissas  cuts  the  locus,  for  these  are  the  real  roots  only,  but 
the  number  of  times  that  any  straight  line  can  cut  the  locus 
includes  the  whole  number  of  roots,  real  and  imaginary. 

EXAMPLES. 

Form  the  equations  having  the  following  roots,  both  by 
Cor.  2  and  Cor.  3. 

1.  5,     3,     3>     —  1,     —  3>     —  5- 

2.  a,     b,     c,     d. 

3.  2,      1  +  V—  3>     1  —  \/Z=~3- 

4.  2,     —  2.      —  3  +  V—  2>      —  3  —  V—  2. 

5.  2  ±  a/^5>     3  ±  V-  7.-     —  1  ±  V—  i- 


NUMERICAL     HIGHER     EQUATIONS.  247 

IV.      FORMS     OF     ROOTS. 
Theorem  III. 

549.  The  imaginary  roots  of  f(x)  =  o  will  be  found  in 
conjugate  pairs. 

That  is,  if  a+  -\/ —b  be  a  root,  a  —  -\/—b  will  also  be  a  root. 

Demonstration. — Since  the  sum  and  the  product  of  the  roots  are 
both  real,  there  cannot  be  an  odd  number  of  imaginary  roots,  nor  an  even 
number,  except  when  found  in  conjugate  pairs  whose  sum  and  product 
are  both  real. 

Cor.  i.  f(x)  will  have  a  real  quadratic  factor  for  each 
pair  of  imaginary  roots. 

For    [x  -  (a  +  \/~b)~]  x  [x  -(a-  \/~b)]  =  (x  -  a?  +  b. 

Cor.  2.  f  {x)  may  be  separated  into  real  factors  ;  of  the  first 
degree,  corresponding  to  each  real  root ;  and  of  the  second  degree, 
corresponding  to  each  p air  of  imaginary  roots. 

Cor.  3. — The  coefficients  off(x)  maybe  so  changed  as  to 
give  two  equal  or  two  unequal  real  roots  in  place  of  each  pair 
of  imaginary  roots. 

For,  the  factors  which  produce  the  quadratic  factor  (x— af  +  b  are 
imaginary  when  b  is  positive,  real  and  equal  when  b  is  zero,  and  real 
and  unequal  when  b  is  negative.  Bat  a  change  of  b  from  positive  to 
negative  through  zero  will  produce  no  change  in  f(x),  except  in  its 
coefficients.  Hence,  the  coefficients  may  be  so  changed  as  to  require 
b  —  o,  or  b  <  o. 

Cor.  4. — The  product  of  the  imaginary  roots  of  f(x)  =  o 
is  always  positive. 

For,     (a  +  \/—  b)  (a  —  \/—  b)  =  ai  +  b,   which  is  always  positive. 
Hence, 

Cor.  5. — In  an  equation  whose  roots  are  all  imaginary,  the 
coefficient  of  x°  will  be  positive. 


248  NUMEEICAL     HIGHER     EQUATIONS. 

Cor.  6. — Every  equation  of  an  odd  degree  has  at  least  one 
real  roof,  opposite  in  sign  to  the  absolute  term,  but  an  equation 
of  an  even  degree  mag  have  all  its  roots  imaginary. 

Cor.  7. — An  equation  of  an  even  degree  toliose  absolute 
term  is  negative,  tuill  have  at  least  two  real  roots,  one  positive 
and  one  negative. 

This  theorem  and  its  corollaries  are  illustrated  by  the  loci  of  equa- 
tions. 

For  example,  Figs.  VII,  VIII,  and  IX  represent  a  locus  of  the  4th 
order  in  different  positions  with  reference  to  the  axis  of  abscissas.  By 
reference  to  equations  (16),  (17),  and  (18)  (Art.  5341,  which  give  those 
different  positions,  it  will  be  observed  that  they  differ  only  in  the  abso- 
lute term,  an  increase  in  this  term  carrying  the  locus  upward  and  a 
decrease  downward.  This  is  evidently  as  it  should  be,  for  a  change  in 
the  absolute  term  produces  a  corresponding  change  in  each  of  the  ordi- 
nates. 

It  appears  therefore  that  a  curve  of  an  even  degree,  going  as  it 
does  to  infinity  in  only  one  direction  and  that  upward,  may  be  so  placed, 
by  giving  a  proper  value  to  the  absolute  term  of  its  equation,  that  the 
axis  of  abscissas  will  not  cut  it  at  all,  as  in  Fig.  IX,  making  all  the  values 
of  x  imaginary  when  y  =  o. 

Or,  as  in  Fig.  VIII,  the  axis  may  cut  the  curve  twice  only,  giving 
two  real  and  two  imaginary  roots.  In  passing  from  the  position  Fig.  IX 
to  Fig.  VIII,  it  would  pass  the  position  Fig.  VII,  in  which  the  roots  are 
all  real,  two  positive  and  two  negative,  the  positive  roots  being  equal, 
and  also  the  negative. 

By  making  the  absolute  term  of  the  equation  i,  a  position  for  the 
locus  would  be  found  in  which  the  roots  would  all  be  real  and  unequal. 
It  appears  also  from  this  illustration  that  when  the  equation  is  so 
changed  as  to  drop  out  a  real  root,  it  must  drop  out  two  such  roots ; 
hence  the  number  of  imaginary  roots  will  be  even. 

Fig.  X  shows  a  curve  of  the  oth  order,  or  one  whose  equation  is  of 
the  5th  degree.  This  curve  goes  to  infinity  both  upward  and  downward, 
as  do  all  curves  whose  equations  are  of  an  odd  degree;  see  Figs.  II,  III, 
V,  and  VI.  These  curves  cannot  be  moved  up  or  down  so  far  that  they 
will  not  be  cut  at  least  once  by  the  axis  of  abscissas.  The  roots  of  their 
equations,  therefore,  cannot  all  be  imaginary. 

In  equation  19  (Fig.  X),  making  the  absolute  term  —25  will  make  all 
the  roots  but  one  imaginary,  and  that  will  be  positive;  making  the 
absolute  term  1  or  any  greater  number  will  leave  but  one  real  root,  and 
that  will  be  negative. 


NUMERICAL     HIGHER     EQUATIONS.  249 


Theorem  IV. 

550.  If  f (x)  has  all  its  roots  imaginary,  it  will  have  a 
positive  value  for  every  real  value  of  x. 

Demonstration. — By  the  supposition,  f(x)  is  the  product  of  factors 
of  the  second  degree,  each  formed  from  a  pair  of  imaginary  roots ;  as, 

(a;  —  a  +  -y/—  b)(x  —  a  —  -y/—  b)  =  (x  —  a)-  +  b. 

Each  of  these  factors  is  +  for  all  values  of  ,r  ;  hence  their  product 
is  +.  Notice  how  this  is  illustrated  hy  Fig.  IX.  The  curve  being  all 
above  the  axis  of  abscissas,  when  the  roots  are  imaginary  the  ordinates 
are  positive  for  all  values  of  x. 

Cor. — The  sign  of  f  (x)  for  any  real  value  of  x  will  depend 
on  the  real  roots. 

Theorem  V. 

551.  The  equation  f(x)  =  o  can  have  no  fractional  root. 

Demonstration. — Let  -  represent  a  fraction  reduced  to  its  lowest 

terms,  and  suppose  this  fraction  to  be  a  root  of  f(x)  =  o ;  then  substi- 
tuting, we  have, 

a"        .   nn~l        .   an~2  . 

j-    +   Ax  -— r   +   A*  =— =    + +   An   =  O. 

b"  b"~l  on~z 

Multiplying  by  b"-\ 

y-  +  A  iffl"-1  +  Aian~2b  +....  +  Anb"-1  =  o. 
b 

All  the  terms  after  the  first  in  this  equation  are  integral,  and  the  first 
term  is  an  irreducible  fraction.  But  the  sum  of  integers  and  an  irre- 
ducible fraction  cannot  be  zero.     Hence  the  last  equation  is  absurd,  and 

-  is  not  a  root  of  f(x)  =  o. 

Cor. — The  real  roots  of  f(x)  =  o  will  be  either  integral  or 
incommensurable,  since  these  are  the  only  real  quantities, 
except  fractions. 

Note. — Incommensurable  Hoots  are  such  as  cannot  be  meas- 
ured with  the  unit  of  measure  employed,  and  therefore  require  an  infinite 
number  of  terms  to  express  them.  Such  a  quantity  is  y/2,  which  gives 
rise  to  an  endless  decimal. 

A  Fraction  is  one  or  more  of  the  finite  parts  of  a  unit,  and  can 
always  be  expressed  by  a  finite  numerator  and  denominator. 


250  NUMERICAL     HIGHEi;      E  Q  U  A  T  J  0  X  S  . 

V.     SIGNS     OF     THE     ROOTS. 

Theorem  VI. 

552.  Changing  the  signs  of  the  alternate  terms  of  f{x)  =  o 
changes  the  signs  of  its  roots. 

Demonstration. — By  Theorem  II,  Cor.  3,  changing  the  signs  of  the 
roots  changes  the  signs  of  the  alternate  terms,  beginning  with  the  term 
containing  xn~ *.  But  changing  the  signs  of  the  alternate  terms,  begin- 
ning with  x",  gives  the  same  equation  by  afterwards  changing  all  the 
signs,  which  does  not  affect  the  roots. 

Or  the  theorem  may  be  proved  as  follows  :  If  +a  and  —a  be  substi- 
tuted for  x  in  /{.'  ,  tin'  only  difference  in  the  result  will  be  opposite 
signs  for  the  terms  containing  the  odd  powers  of  x.  Hence  if  a  lie  a  root 
of  f(x)  =  0,  —a  will  be  a  rout  of  the  same  equation  with  the  signs  of  the 
odd  powers  of  x  changed,  or  with  the  signs  of  the  even  powers  changed, 
since  this  will  give  the  same  result. 

Theorem  VII. 

553.  In  f{x)  =  o,  the  number  of  positive  roots  cannot  be 
greater  than  the  number  of  variations  if  sign,  nor  the  number 
of  negative  roots  greater  than  the  permanences. 

Note. — When  two  consecutive  signs  in  an  equation  are  alike,  it  is 
called  a  Permanence  ;  when  unlike,  a  Variation. 

Demonstration. — Let  the  signs  of  f(x),  taken  in  their  order,  be 
+  —  ++  —  +.     If  now  a  new  positive  root  (a)  be  introduced,  f(x)  will 

be  multiplied  by  x— a.     This  multiplication  will  give,  using  only  the 
signs, 

+   —   H-    +   —    + 
+   — 


+   —   ++   —   + 

—   + +   — 

+   —    +    ±    —   +   — 


from  which  we  see  that  using  either  sign,  where  there  is  an  ambiguity 
we  have  added  a  variation.  In  like  manner,  multiplying  by  +  +  will 
add  a  permanence. 


NUMERICAL     HIGHER     EQUATIONS.  251 

Cor.  i. — If  the  roots  of  an  equation  be  all  real,  the  number 
of  positive  roots  ivill  equal  the  number  of  variations,  and  the 
number  of  negative  roots  the  number  of  permanences. 

For,  in  that  case  tlie  whole  number  of  roots  will  equal  the  whole 
number  of  permanences  and  variations  together ;  hence  by  the  theorem 
the  corollary  must  be  true. 

Cor.  2. — If  an  equation  be  incomplete,  and  the  signs  before 
and  after  the  missing  term  or  terms  be  alike,  the  equation  must 
have  imaginary  roots. 

For  the  intervening  terms  whose  coefficients  are  zero  may  be  either  + 
or  — ,  and  we  shall  have,  when  there  is  one  missing  term,  +  ±  +  or  — 
±  — ,  and  in  either  case  we  may  count  either  two  permanences  or  two 
variations. 

But  the  positive  roots  cannot  be  greater  than  the  least  number  of 
variations,  nor  the  negative  roots  greater  than  the  least  number  of 
permanences ;  hence  the  two  cannot  be  equal  to  the  degree  of  the 
equation. 

For  example,  in  the  equation 

Xs  +  33J3  —  x  +  7  =  o, 

we  shall  have  the  signs 

+      ±      +      ±     —      +. 

From  which,  counting  as  few  variations  as  possible,  we  have  2  ;  hence 
there  cannot  be  more  than  2  positive  roots.  Counting  as  few  perma- 
nences as  possible  we  have  i,  and  there  cannot  be  more  than  i  negative 
root.  But  there  are  5  roots  in  all,  hence  there  must  be  at  least 
2  imaginary  roots. 

Cor.  3. — If  two  or  more  consecutive  terms  are  wanting, 
the  equation  will  have  imaginary  roots  whatever  may  be  the 
signs  of  the  preceding  and  following  terms. 

Eow  many  positive  and  how  many  negative  roots  can  the 
following  equations  have,  and  how  many  of  their  roots  must 
be  imaginary  ? 

1.  .r7  —  514  +  6x2  -  5/  I  1  =  o, 

2.  x6  —  2a*2  +  5=0. 

3.  X5  +  X  +1=0. 

4.  x5  —  x2  —  5  =0, 


252  NUMERICAL     HIGHER     EQUATIONS. 


VI.     LIMITS    OF    ROOTS. 

554.  A  Superior  Limit  to  the  roots  of  an  equation, 
is  a  number  known  to  be  larger  than  the  largest  root. 

555.  An  Inferior  Limit  is  a  number  known  to  be 

less  than  the  least  root. 

It  is  sometimes  convenient  to  find  such  limits  which  shall  confine 
the  search  for  roots  within  as  narrow  bounds  as  possible.  This  may  be 
done  by  the  following  theorems : 


Theorem    VIII. 

556.  If,  in  f(x)  =  o,  two  different  numbers  (not  roots) 
be  substituted  for  x,  the  signs  of  the  result  will  be  alike  when 
there  is  an  even,  and  unlike  when  there  is  an  odd  number  of 
real  roots  situated  between  the  numbers  substituted. 

Demonstration. — Let 

(as— a^  {.r—a,)  (x-a3)  .  .  .  (x—an-j,)  (i) 

(p  being  the  number  of  imaginary  roots)  be  the  product  of  the  factors  of 
f\.r),  which  are  formed  from  the  real  roots  of  f(x)  =  o.  The  sign  of 
this  product  tor  any  value  of  x  will  be  the  same  as  of  /(.n  for  the  same 
value  of  x ;  for  the  product  of  the  imaginary  factors  is  always  +. 
(Art.  550). 

Let  it  be  supposed  that  the  roots  alt  a.,,  aA,  etc.,  are  in  the  order  of 
their  magnitude,  ax  being  greatest,  a%  the  next,  and  so  on. 

If,  now,  a  number  {a')  greater  than  a,  be  substituted  for  x,  the  factors 
of  (1)  will  all  be  positive,  and  the  result  will  be  positive  ;  but  if  a 
number  {a")  less  than  ",  and  greater  than  a.2  be  substituted  for  .r,  there 
will  be  one  negative  factor,  and  the  rest  will  be  positive.  The  result  will 
therefore  be  negaticc. 

In  like  manner  if  the  number  substituted  be  diminished  till  it  is  lrss 
than  as,  the  result  will  change  its  sign  again  and  become  positive. 
Thus  we  see  that  for  every  root  which  is  passed  in  diminishing 
the  number  substituted,  the  sign  of  the  result  changes,  an  even  number 
of  changes  giving  like  signs  and  an  odd  number  unlike.  If  there  be  two 
or  more  equal  roots  they  will  be  passed  at  the  same  time,  but  this  does 
not  affect  the  truth  of  the  theorem. 


NUMERICAL     HIGHER     EQUATIONS.  253 

This  theorem  is  also  illustrated  by  the  loci  of  equations.  Referring 
to  the  preceding  chapter  it  will  be  observed  that  the  ordinate  of  any 
point  of  a  locus  is  the  value  obtained  by  substituting  the  value  of  the 
abscissa  for  x  in  /(«).  By  taking  any  two  abscissas,  it  will  be  seen  that 
the  corresponding  ordinates  have  opposite  signs  when  the  number  of 
intermediate  roots  is  odd,  and  the  same  sign  when  that  number  is  even. 

Note. — It  must  not  be  forgotten  that  zero  is  an  even  number. 

Cor. — If  a  number  less  than  the  least  root  of  f(x)  =  o 
be  substituted  for  x,  the  result  will  be  positive  when  the 
equation  has  an  even  number  of  real  roots,  and  negative  ivhen 
an  odd  number. 

Theorem    IX. 

557.   If  Am  be  the  largest,  and  Ah  the  first  negative  coeffi- 

dent  of  f(x)  =  o,   then   {Am)h  +  i    is  a  superior  limit  to 
its  roots. 

Demonstration. — Any  value  of  x  which  renders  f(x)  positive,  and 
of  which  it  can  be  shown  that  all  greater  values  will  render  it  positive, 
is  a  superior  limit  to  the  roots.  Suppose  all  the  terms  after  the  first 
negative  term  to  be  negative,  and  the  coefficients  of  the  negative  terms 
each  to  be  equal  to  Am-  As  this  will  be  the  most  unfavorable  case  pos- 
sible, if  we  can  show  that  when 

i 
x  =  or  >  (A,,,)*  +  i,  (i) 

then  x"  >  Am  (.i:"-ft  +  as*-*-1   .  .  .  x  +  i),  (2) 

the  theorem  will  be  proved,  forf(x)  will  then  be  positive. 

Subtracting  1  from  each  member  of  (1)  and  raising  it  to  the 
ht!l  power, 

(<E  —  i)>>  =  or  >  Am, 

or  (,r  —  i)*-1  (x  —  1)  =  or  >  Am  ; 

Xh~l  (X  —  I)  >  Am, 
x-(h-\) 

and  1  >  Am . 

x  —  1 

Multiplying  by  xn, 

xn  >  A 

or  xn  >  A 


-  1 

X>'~h+\  —  1 


254  -\  r  m  e  r i c  a l    ii  i  <;  he:;    e y  u  a  t  i  o  x  b . 

Making  the  division, 

X»   >    Am  (X'~!>   +  X"-^-1    .    .    .    X  +    i). 

1 

Hence,  (Am)h  +  i  is  a  superior  limit  to  the  roots  off(x)  =  o,  A,„  hi  inn 
the  greatest  negative  coefficient  and  h  the  number  of  terms  preceding  t/w 
first  negative  coefficient. 

Cor. — //"  the  signs  of  the  alternate  terms  of  an  equation  he 
changed,  a  superior  limit  to  the  roots  of  the  resulting  equation 
will,  by  changing  its  sign,  become  an  inferior  Until  to  the  roots 
of  the  original  equation.     (Theorem  VI.) 


VII.     LIMITING    EQUATION. 

558.  In  the  following  discussions,,  /'(•'')  represents  the 
first  differential  coefficient  of  fix),  and  0  (.< )  the  greatest 
common  divisor  of  f{x)   and  f'(x). 

Theorem    X. 

559.  The  real  roots  of  f  (x)  =  o  are  situated  between 
those  of  f{x)  =  o;  that  is,  if  at,  az,  a3,  etc.,  are  roots  of 
_/'(./•)  =  o,  then  f'(x)  —  o  has  a  root  between  at  and  a», 
also  between    a2    and  az,   and  so  on. 

Demonstration. — Suppose  tlie  roots  of  f(.r)  to  be  real.  Then 
Vve  have, 

/(.?■)  =  (x—ax)  (x—az)  (jj— a8)  ....  (x-a,,). 

f(x)  =  (x—a2)  (x-o.,)  (<c— a4)  ....  ix-a,,)  + 
(x—aA{x-aB)(x—aA  ....  {x—a„)  + 
(x— ax)  (x—a2)  (x— a4)  ....  (as—a,,)  + 


(x-aj  (x-az)  (x-as)  ....  (x-a„-\), 

in  which  ax,  n.,,  as,  etc.,  arc  the  roots  of  f(x)  —  o,  in  the  order  of  mag- 
nitude, ai  being  the  largest.     (Art.  414.  Rule  III.) 

Any  number  equal  to  or  greater  than  ",,  substituted  for  x  in  j  {x), 
will  render  each  factor  of  each  term  positive,  and  therefore  the  whole 
function  positive;  a1  is  therefore  a  superior  limit  to  the  roots  of/'  (x)=0. 


NUMERICAL     HIGHER     EQUATIONS.  255 

If  a„  be  substituted,  all  the  terms  but  the  second  will  become  zero,  and 
that  will  have  one  negative  factor  and  the  function  will  therefore  be 
negative. 

So  also  a3  substituted  for  x  will  make  the  function  positive,  and  the 
successive  substitutions  of  alf  a.,,  a3,  etc.,  will  give  alternately  positive 
and  negative  values  for  f'(x)-  Hence,  (Theorem  VIII)  f'{x)  —  o  has  an 
odd  number  of  roots  between  each  two  consecutive  roots  of  f(,v)  =  o. 
This  number  cannot  be  less  than  one. 

If  f(.r)  =  o  has  imaginary  roots,  the  proof  is  not  affected,  since  the 
sign  of  f'(x)   depends  wholly  upon  its  real  roots. 

560.  The  equation  /'(%)  =  o  is  called  the  Limiting 
or  Separating  Equation,  on  account  of  the  situation  of 
its  roots  between  those  of  f(x)  =  o. 

Referring  to  Fig.  X,  Art,  533,  we  see  that,  when  the  equa- 
tion of  this  locus  is  so  changed  in  its  absolute  term  as  to  give 
two  equal  roots,  the  axis  of  abscissas  will  pass  through  one  of 
the  points  m,  m',  m",  or  m'",  and  the  equal  roots  will  be  the 
abscissa  of  one  of  these  points.  At  the  same  time  this  abscissa 
will  be  a  root  of  /'(%)  =  °>  since  one  of  its  roots  is  between 
the  two  equal  roots  of  f(x)  =  o.  But  the  change  in  the 
absolute  term  of  the  equation  of  the  locus,  which  caused  the 
axis  of  abscissas  to  pass  through  m,  makes  no  change  in  f'(x). 
(Art.  413,  Rule  II.)     Hence, 

The  roots  of  f'(x)  =  o  are  the  abscissas  of  the  points  m, 
m',  etc. 

VIII.     EQUAL    ROOTS. 
Theorem    XL 

561.  If  f(x)  =  o  has  m  roots  each  equal  to  a,  f  (x)  =  o 
will  have  m  —  i  such  roots ;  and  (x  —  a)m~l  will  be  a 
common  divisor  of  f(x)   and  f  (x). 

Demonstration.— This  is  evident  from  Theorem  X,  for  if  any  two 
roots  of  /(.r)  =  o  become  equal,  the  root  of  f'(x)  =  o  between  them 
must  be  the  same,  and  if  m  roots  of  f(x)  =  o  become  equal,  m  —  1  roots 
of  f'(.r)  —  o  must  also  be  the  same.  This  will  obviously  give  (x—a)"'~l 
as  a  common  divisor  of  f(x)  and  f'(x).  This  common  divisor  will  have 
other  factors  if  f(x)  has  other  equal  roots,  and  not  otherwise, 


256  NUMERICAL     HIGHER     EQUATIONS. 

This  theorem  may  aJso  be  proved  by  reference  to  the  form  of  f{x) 
and  f'(x),  under  Theorem  X. 

A  comparison  of  these  functions  shows:  That  if  alt  a.,,  a3,  etc.,  are 
all  unequal,  tbere  is  no  common  divisor  of  the  two,  for  each  factor  of 
f(.r)  is  wanting  in  some  one  term  of  /   ./ :). 

But  if  any  two  of  the  roots  al,  a.,,  aa,  etc.,  are  equal,  making  two  of 
the  factors  of  j\x)  equal,  then  that  factor  will  be  found  in  every  term 
of  f'(x),  an(l  "iU  therefore  be  a  common  divisor  of  the  two  functions. 
So  if  three  of  the  factors  of  f(.r)  are  equal,  that  factor  would  be  found 
twice  in  f\x),  and  therefore  twice  in  the  common  divisor.  Also  if  there 
be  two  sets  of  equal  factors,  the  same  would  be  true  of  each. 

Cor.  i. — If  f(x)  =  o  has  no  equal  roots,  there  is  no  com- 
mon divisor  of  f(x)  and  f'(x). 

Cor.  2. — If  (p  (x)  =  o  has  m  roots  each  equal  to  a, 
f(x)  =  o  will  have  w,  and  f{x)  =  o,  m  +  i  such 
roots. 

Cor.  3. — If  4>(x)  =  o  has  no  equal  roots,  f(x)  is 
divisible  by  (p(x)  twice,  and  each  of  the  roots  of  <p(x)  =  o 
is  found  twice  as  a  root  of  f(x)  =  o. 

562.  This  theorem  furnishes  the  means  of  forming  from 
any  equation  having  equal  roots,  two  or  more  equations  whose 
roots  shall  be  the  roots  of  the  given  equation. 

These  equations  are  formed  by  putting  (p(x)  equal  to  zero, 
and  also  the  quotient  found  by  dividing  f(x)  by  <p{x). 

In  this  way  an  equation  of  a  higher  degree  may  often  be 
separated  into  factors  and  reduced.  Take  for  illustration  the 
following  : 

1.  f(x)  =  xh  —  15a"3  +  io.r2  +  6o.c  —  72  =  o. 

We  find  f'{x)  =  sx4  —  45.J2  +  20.r  +  60. 

From  this  we  may  throw  out  the  factor  5,  giving 

x4  —  qx*  +  4.7-  +  12. 

Finding  the  g.  c,  (I.  of  these  two  functions,  wo  have 

<p(x)  =  ,r3  —  ,r-  -  Sx  +  12. 


NUMERICAL     HIGHER     EQUATIONS.  257 

Testing   this  again   for   equal   roots   by   trying   for   a   common  divisor 
of  (}>(x),  and 

<j>'(x)  =  3X'2  —  2.i'  —  8, 

we  find  that  divisor  to  be   x  —  2. 

Putting  x  —  2  =  o, 

we  have  x  =  2, 

one  of  the  roots  of  <p(x)  =  o. 

Dividing   <j>  (x)  =  o  by   x  —  2,  we  have 

x-  +  x  —  6  =  o. 
Fi'om  this  we  get 


x  =  —  i  ±  y^6  +  i  =  —  £  ±f  =  2  or  —  3. 
We  have  now  the  three  roots  of  <p  (x)  =  o,  viz. : 
2,         2,         and         —  3. 
We  might  now  divide  f(.v)  —  o  by   <p  (x),  and  we  should  find 

cc2  +  x  —  6  =  o, 

which  would  give  us  the  roots  2  and  —3.  But  we  need  not  make  this 
division,  since  when  we  have  the  roots  of  (j>(x)  =  o,  viz. :  2,  2,  and  —3, 
we  know  that  2,  which  is  twice  a  root  of  <p  (x)  =  o,  is  three  times  a 
root  of  f(.r)  =  o  ;  and  —3,  which  is  once  a  root  of  <j>(x)  =  o,  is  twice 
a  root  of  f(x)  =  o.  ( Theorem  XI,  Cor.  2.)  Hence  the  roots  of  f(x)  =  o 
are  known  to  be  2,  2,  2,  —3,  and  —3,  as  soon  as  we  have  the  roots  of 
<?(.*')  =  o. 

Iu  like  manner  find  the  roots  of 

2.  Xs — 6x~  +  8a;6 4-  1  Sx5 — $-jxi-\-$6x3-{-$2x2 — 48^+16  =  o. 

3.  x5  —  x4  -f  4X3  —  4a;2  4-  \x  —  4  =  0. 

4.  x*  —  io./'3  +  24X2  4-  lox  —  25  =  o. 

5.  x5  4-  2XA  —  t,x3  —  3^2  4-  2X  4-1=0. 

563.  If  an  equation  has  roots  numerically  equal,  but  with 
opposite  signs,  changing  the  signs  of  the  alternate  terms  will 
give  an  equation  having  the  same  numerically  equal  roots, 
but  with  its  other  signs  different.  Hence,  the  greatest 
common  divisor  of  the  two  equations  will  have  these  equal 
root     and  no  others. 


258  NTUMEEICAL     HIGHEE     EQUATIONS. 


IX.     COMMENSURABLE     ROOTS. 

564.   The  Commensurable  Roots  of  /(■>■)  =  o  are 

all  integral  (Art.  551,  Theorem  V),  and  may  be  found  by  the 
following-  theorem : 

Theorem    XII. 

565.  If  ar  be  a  root  of  f{x)  =  o,  and  A„  be  divided  by 
a,  and  An_x  added  to  the  quotient,  and  if  this  sum  be  divided 
by  a,,  and  An-2  added  to  the  quotient,  and  /I/is  process  be 
continued  till  all  tlie  coefficients  of  /(.r)  have  been  used,  the 
result  will  be  zero. 

Demonstration. — A  careful  consideration  of  the  manner  in  which 
the  coefficients  are  formed  from  the  roots,  will  make  the  truth  of  the 
theorem  plain.     An  is  the  product  of  the  roots  with  their  signs  changed, 

.-. —  the  product  of  all  the  roots,  except  ar,  trittt  their  signs  changed. 

A„-\  is  the  sum  of  the  products  of  the  roots,  with  their  signs  changed, 
taken  n  —  1  at  a  time.     Each  of  the  terms  of  A„-\   will  therefore  be 

divisible  by  ar  except  one,  and  that  term  will  equal  — '  ".     Therefore 

adding  — -  to  A„-i,  cancels  the  onlv  term  which  does  not  contain  the 

ar 

factor  ar,  and  makes  A„-\  divisible  by  ar. 

In  like  manner,  An  2  is  the  sum  of  the  products  of  the  roots,  with 
their  signs  changed,  taken  n— 2  at  a  time,  and  the  terms  not  containing 
ar  will  equal  the  last  quotient  obtained,  with  its  sign  changed.  Hence, 
adding  that  quotient  to  An-2  cancels  the  terms  not  containing  the  factor 
ar,  and  renders  A„  -2  divisible  by  ar. 

In  the  same  way  each  coefficient  is  made  divisible  by  a ...  But  .1, 
is  the  sum  of  the  roots,  and  when  the  terms  not  containing  the 
factor  ar  are  cancelled,  there  will  remain  —ar,  which  divided  by  ar  gives 
—  r,  and  this  added  to  1,  the  coefficient  <»!'  .<■■,  i;'ives  zero. 

566.  To  find  the  integral  roots  of  an  equation  by  the 
application  of  this  theorem,  use  the  following 

Rule.  — I.  Write  in  a  line  the  integral  factors  of  the 
absolute  term  (A„). 


NUMERICAL     HIGHER     EQUATIONS.  259 

II.  Divide  A„  by  each  of  these  factors,  and  write  each 
quotient  below  its  divisor. 

III.  Add  An-\  to  each  quotient  and  write  the  sum 
below. 

IV.  Divide  each  sum  by  that  factor  of  An  which  stands 
above  it,  and  continue  in  like  manner  to  add  the  successive 
coefficients  and  to  divide,  until  the  coefficients  are  all  used. 

V.  If,  in  the  course  of  the  operation,  the  division  by  any 
one  of  the  factors  is  imperfect,  that  factor  is  not  a  root  and  the 
ivorh  with  it  will  cease. 

To  illustrate  the  rule  take  Ex.  2.  (Art.  562.)  The  opera- 
tion will  be  as  follows  : 

Divisors,  1,        2,        4,        8,      16,    — 1,    —2,    —4,    —8,  — 16,  Fac.of  A„. 
16,        8,        4,        2,        1, —16,    —8,    —4,    —2,    —1, Quotients. 
Add     —48  =  An— 1. 


Add 

-32,  -40,  -44,  -46, 
—  32,  —20,  —11, 

32  =  An-2. 

-47.  -64. 
64, 

-56, 

2S, 

-52, 
13 

—  50,  —49,  Sums. 

Quotients. 

Add 

O,        12,    +21, 
0,         6, 
36  =  An-3. 

96, 
-96, 

60, 
-3D, 

45, 

Sums. 
Quotients. 

Add 

36,        42, 

36,        21, 

—  57  =  An-i. 

-6o, 
60, 

6, 
-3. 

Sums. 
Quotients. 

Add 

—  21,   —36, 
-21,  -l8, 
18   =  An-5- 

3, 
-3, 

-60, 
30, 

Sums. 
Quotients. 

Add 

-3,       0, 
-3,        0, 
8  =  An-6. 

15, 
-15, 

48, 
-24, 

Sums. 
Quotients. 

Add 

5,        8, 

5.        4, 

-6  =  A,,--. 

-7, 
+  7, 

-16, 

8, 

Sums. 
Quotients. 

—  1.     —2, 

—  1,    —1, 

+  1, 
—  1, 

2, 
—  1, 

Sums. 
Quotients. 

260  NUMERICAL     HIGHER     EQUATIONS. 

The  work  need  be  carried  no  farther  to  show  that  the  integral  roots 
are  i,  2,  —  1  and  —  2.  This  does  not  determine  whether  these  roots  are 
once  or  more  than  once  roots  of  the  equation,  but  the  first  member  may 
now  be  divided  by  the  factors  x— 1,  .1—2,  x+i  and  x+2,  and  the 
quotient  put  equal  to  zero  will  be  an  equation  whose  roots  will  be  the 
other  four  roots  of  the  primitive  equation. 

Making  this  division  by  the  synthetic  method  as  follows  : 

1  —  6  +     8  +  18  -  57  +  36  +  32  -  48  +  16  I  2 
2  —    8  +     o  +  36  —  42  —  12  +  40  —  16 

o  ]  —2 


I  -  4  + 

0  +  18  —  21  — 

6  + 

20  —  8  +  c 

—  2  + 

12  —  24  +  12  + 

18  — 

24+8 

i  -  6  + 

12  —  6  —  g  + 

12  — 

4  +  0  j_£ 

+  1  — 

5  +  7  +  1  — 

8  + 

4 

1  -5  + 

7  4-  1  —  8  + 

4  + 

0  ,  -1 

—  1  + 

6  —  13  +  12  — 

4 

1— 6+13  —  12+     4+     o 
We  have  for  the  equation  having  the  other  four  roots, 

X4  —  6x3  +    13a?  —   123!   +   4    =    0. 

Applying  the  rule  for  integral  roots, 

4,       —1,       -2,       -4, 
x»       —4.       —2,       —1, 


I, 

2, 

4. 

2, 

—  12, 

-8, 

—  10, 

— 11, 

-16, 

—14, 

-13, 

-8, 

-5, 

16, 

7, 

13, 

5. 

8, 

29, 

20, 

5, 

4, 

-29, 

-10, 

-6, 

—  1, 

—  2, 

-35, 

-16, 

—1, 

—  1, 

35- 

8. 

We  find  1  and  2  are  roots  of  the  equation. 
Dividing  out  these  roots  gives 

x2  —  3.?'  +  2  =  0, 

whose  roots  are  r  and  2.     Hence  the  roots  of  the  primitive  equation  are 
2,     2,     2,     1,     1,     1,     —2,     and     —1. 


NUMERICAL     HIGHER     EQUATIONS.  261 

567.    Let  the  student  now  apply  the   principles  already 
developed  to  the  following 


EXAMPLES. 

Each  of  the  following  questions  should  be  answered  in 
reference  to  the  equations  below : 

i.  How  many  roots  has  the  equation  ?     (Theorem  II.) 

2.  What     is     their      product     and     what     their     sum  ? 
(Theorem  II,  Cor.  3.) 

3.  What  numbers  may  be  its  integral  roots  ?    (Theorem  I, 
Cor.  2.) 

4.  What  limits  to  its  roots  can  be  fixed  ?     (Theorem  IX.) 

5.  Has  it  an  even   or  an   odd   number  of    real    roots? 
(Theorems  II  and  III.) 

6.  Has  it  an  even  or  an  odd  number  of  negative  roots? 
(Art.  547,  Cor.  4.) 

7.  Has  it  an  even  or  an  odd  number  of  positive  roots  ? 
(Art.  547,  Cor.  4.) 

8.  How  many  positive  and  how  many  negative  roots  can 
the  equation  have  ?     (Theorem  VII.) 

9.  What  are  its  commensurable  roots  ?     (Theorem  XII.) 

10.  What  is  the  value  of  the  first  member  when  5  is  substi- 
tuted for  x  ?     (Theorem  I.) 

1 1.  What  equation  has  the  same  roots  with  opposite  signs  ? 
(Theorem  VI. ) 

12.  What    equation     has    its    roots    twice     as    large  ? 
(Art.  542.) 

13.  What  equation  has  intermediate  roots  ?   (Theorem  X.) 

1.  x6  +  45a:4  —  246a;2  +  200  =  o. 

2.  .t4  +  $x3  —  2X  —  156  =  o. 

3.  z6  —  sx*  4-  3.r3  +  6x*  —  297  =  o. 

4.  x5  +  9Z4  +  7-r3  —  3a;2  —  4X  +  10  =  o. 


-MI-.' 


N   I    U  i:  K  I  <    A  L      II  I  ( I  II  E  It      E  < >  D  A  T  I  0  ij  S 


5-  X5  —  22T4  +  3-r3  —  3^2  4-   2X  —  1    =0. 

6.         x~'  —  5a-'6  +  3X4  +  9a;2  +  12  =  o. 

Transform    the    following    to    the    form    (1)    and    give 
the  changes  produced  in  the  roots  by   the  transformation. 

(Arts.  538,  542.) 


7- 
8. 

9- 
10. 

1 1. 


x%  _  g^t  _|_  >j£s  -f.  *x*  _  1  —  o. 

3^5  -[_  ya;o  —  6.rs  +  2./  —  2=0. 

111 

2X-  -{-  3X*  +  4£T  —  2.r  +2  =  0. 

9^  +  32    4  —  2£    a    4-   2    =   O. 

a;2  —  a;       x5  —  a:2       a^  —  x 


Remove  the  equal  roots  from  the  following :    (Theorem  XL) 

12.  x5  +  a4  —  4a;3  —  4a:2  4-  42;  +  4  =  o. 

13.  xG  —  Sx5  +  26a;4  —  44a:3  +  41a;2  —  2o.r  +  4  =  0. 

14.  a;6  —  3a.4  —  45a;2  —  81  =  o. 

Produce  equations  having  the  following  roots: 


h     —2,     3- 

3>     4,     V2,     —V*- 

—5>     V— 2,     —  V— 2. 

—  1  +  V—3,     —  1  —  V— 3,     i     2. 

1  ±  V—3,     3  ±  V— 2. 


I 

>• 

2 

7 

"S> 

2, 

I 

"25 

rZr2, 

I, 

2, 

2, 

2. 

V- 


2, 


\/2,       —  A/2. 


NUMERICAL     HIGHER     EQUATIONS.  263 


X.     INCOMMENSURABLE    ROOTS. 

568.  The  process  of  finding  the  Incommensurable 

Hoots  of  an  equation  depends  on  a  theorem  called,  from  its 
discoverer,  "Sturm's  Theorem."  * 

This  theorem  assumes  that  the  equation  has  no  equal  roots  ;  but  as 
we  have  already  seen  how  the  equal  roots  may  be  removed,  we  may 
prepare  any  equation  for  the  application  of  the  theorem. 

569.  Assuming,  then,  that  f  (x)  has  no  equal  roots,  and 
forming  f'(x),  divide  f(x)  by  f'(x),  and  represent  the 
remainder  with  its  signs  changed  by  f)l.2.(x)- 

In  like  manner  divide  f'(x)  by  /„_2  (x),  and  repre- 
sent the  remainder  with  its  signs  changed  by  /„_3  (.r),  and 
proceed  in  the  same  manner  until  a  remainder,  f0  (x),  is 
found. 

Notes. — i.  The  subscripts  n  —  2,  n  —  3,  .  .  .  o,  indicate  the  degree 
of  the  function.  f0  (.r)  is  therefore  independent  of  X,  and  such  a  remain- 
der will  be  found  ;  for,  each  division  may  be  continued  until  the  remain- 
der is  a  unit  lower  in  degree  than  the  divisor,  and  the  division  will  at  no 
time  be  complete,  since  f(x)  has  no  equal  roots.    (Theorem  XI.) 

2.  The  use  of  these  functions  is  such  as  not  to  forbid  multiplying  or 
dividing  them  by  any  positive  numerical  factor.  They  may  therefore  be 
simplified  by  rejecting  all  such  factors,  and  to  avoid  fractions  in  dividing, 
any  dividend  or  divisor  may  be  multiplied  or  divided  by  any  positive 
a  umber. 

570.  The  functions  /ra_2  (x),  /w_3  (x),  etc.,  are  called  the 
Sturmian  Functions,  and  with  f(x)  and  f'(x)  con- 
stitute the  functions  to  which  Sturm's  theorem  relates. 

The  relations  of  these  functions  to  each  other  is  expressed 
in  the  following  equations,  in  which  Qx,  Q2,  etc.,  are  the 
successive  quotients. 

*  See  page  307,  Note  5. 


264  STURM'S     THE  OB  EM. 

f{x)  =  /'(*)x  ft -/_(*).  (i) 

/'  <*)  =  /_  (x)  Xft-  /„_a  (*).  (2) 

/_  (z)  =  /_  (*)  x  Q3  -/„_4  (.«■).  (3) 


/,(*)  =  /i(*)X  &-,-/.(*).  (4) 

571.    Consecutive  Functions  are  those  adjacent,  in 
the  order  /(z),  /'  (as),  /_  (.r),  /_(a:) /o(»). 


Theorem  XIII. 

572.  JVo  /wo  consecutive  functions  can  become  zero  for  ttie 
same  value  of  x. 

Demonstration. — Suppose  that /„_._.  (.n  and  fn-s{x)  could  became 
zero  for  the  same  value  of  x.  Then  by  Kq  nation  (2),  /'  (;r)  =  o,  and  by- 
Equation  (i),  f(x)  =  o.  But  /  (,r)  and  f(x)  cannot  become  zero  for  the 
same  value  of  x .  (Art.  559.)  .\  fn-i  (%)  and/„_3(.r)  cannot  be  zero  at 
the  same  time.  The  same  may  be  proved  of  any  two  consecutive 
functions. 

Cor. — If  f  (x)  or  any  one  of  the  Sturmian  functions 
reduces  to  zero  for  any  value  of  x,  the  adjacent  functions  have 
opposite  signs  for  the  same  value  of  x. 

If  /„_2(.r)  =  o,  in  Equation  (2),  we  have 

f'(X)  =    ~fn-B(X), 

and  in  like  manner  for  any  other  functions. 


XI.      STURM'S      THEOREM. 

573.    If   in  f{x).  f  (x),  /„_,  (x),  /„_3  (x) /,  (x),   two 

different  numbers  l»  substituted  for  x,  and  the  signs  of  the 
resulting  values  of  the  functions  for  each  substitution  be  set 
separately  in  the  order  of  the  functions,  the  difference  in  the 
number  of  variations  in  I  In1  two  cases  will  be  equal  to  the 
number  of  real  roots  of  f(x)  =  o  situated  between  th  numbers 
substituted. 


STURM'S     THEOREM.  2G5 

Demonstration. — ist.  If  the  number  substituted  for  x  be  supposed 
to  change  from  one  value  to  another,  so  as  to  pass  through  all  inter- 
mediate values,  the  several  functions  will  change  their  values  in  a  simi- 
lar manner,  and  whenever  the  value  of  any  function  passes  from  +  to  — 
or  from  —  to  + ,  it  will  pass  through  zero. 

2d.  When  any  intermediate  function  becomes  zero  for  any  value  of  x, 
the  adjacent  functions  have  opposite  signs  for  the  same  value  of  x. 

j  Hence  a  change  of  sign  in  any  intermediate  function  (that  is,  in  any 
function  except  the  first  and  last)  can  have  no  effect  on  the  number  of 

:  variations.     This  is  obvious,  for  the  variations  of   +   +  —  and  +  —  — 

;  are  the  same. 

3d.  As  none  of  the  intermediate  functions  can  affect  the  number  of 

!  variations  by  any  change  that  may  occur  in  their  signs,  and  as  the  last 
function,  being  independent  of  x,  never  changes  its  sign,  any  change  in 

j  the  number  of  variations  must  be  produced  by  a  change  of  sign  in  f(x). 
But  f(x)  will  change  its  sign  whenever  the  number  substituted  passes  a 
root  of  f(x)  =  0.  If  a  number  greater  than  «i  be  substituted,  f(x)  and 
/'  (.c)  will  both  be  positive  and  will  form  a  permanence.  If  now  this 
number  be  supposed  to  decrease  when  it  passes  «i  ,  f(x)  will  change  its 
sign,  while  f  \x)  will  remain  positive,  since  it  has  no  root  so  great  as  ct\ . 
This  change  of  sign  will  make  the  first  two  signs  —  + ,  giving  in 
place  of  a  permanence  a  variation,  and  the  number  of  variations  will  be 
increased  by  one.  As  the  value  of  x  continues  to  decrease,  it  will  pass  a 
root  of  /'  (a;)  =  o  before  it  comes  to  «a  (Th.  X),  and  therefore  f'x  will 
change  its  sign  before  f{x)  changes  again.     This  will  make  the  first  two 

signs ,  but  without  affecting  the  variations.     When  therefore  the 

value  of  x  passes  ai ,  fix)  will  become  positive  and  the  first  two  signs 
will  become  +  — ,  adding  another  variation.  In  the  same  way  a  varia- 
tion will  be  added  whenever  x  passes  a  root  of  f(x)  —  o.  Hence,  the 
theorem  is  proved. 

574.  Sturm's  TJicorcm  is  applied  to  finding  the 
situation  of  the  incommensurable  roots  of  an  equation.  It 
may  also  be  used  to  find  the  commensurable  roots,  but  the 
methods  already  given  are  sufficient  to  determine  these. 

A  single  example  will  illustrate  its  application. 

1.  Find  the  situation  of  the  real  roots  of 
x3  —  3a;2  +  6x  —  5=0. 

SOLUTION. 

f(x)  =  X3  -  3.^  +  635—5. 
/'  (x)  -f-  3  =  a2  —  2a!  +  2.     (Art.  569,  Note  2.) 

/,  (X)  =    —  2X  +  3. 

f0(x)  =  -  I. 


2GG  STURM'S     THEOREM. 

Substituting  in  these  functions  different  values  for  x,  we  have  the 
signs  as  follows  : 

fix).  /'(*).  fi(x).  fo(x). 

+  oo     gives         +  +             —  —  one  variation, 

o         "            —  +             +  —  two  variations. 

—  00         "              —  +               +  —  two 

There  is  therefore  one  real  root  between  +  co  and  o,  and  no  root 
between  —  co  and  o.  That  is,  tbere  is  one  positive  and  no  negative  root. 
Hence,  two  of  the  roots  are  imaginary. 

To  find  the  situation  of  the  real  root,  we  substitute  as  follows : 

1  gives         —  +  +  —         two  variations. 

2  "  +  +  —  —         one  variation. 

The  root  is  therefore  between  i  and  2,  or  i  +  a  decimal. 

We  may  find  more  exactly  the  situation  of  this  root  by  substituting 
i.i,  1.2,  1.3,  etc.,  as  follows- 


I.I 

gives 

— 

+ 

+ 

— 

two  variations 

1.2 

" 

— 

+ 

+ 

— 

two        " 

1-3 

" 

— 

+ 

+ 

— 

two 

1.4 

" 

+ 

+ 

+ 

— 

one  variation. 

Hence  the  root  is  between  1.3  and  1.4 ;  that  is,  it  is  1.3  +  . 

In  the  same  way  other  figures  of  the   root  could  be  found,  but  the 
substitutions  would  be  tedious,  and  we  shall  hereafter  show  an  easier ' 
method  of  carrying  out  the  work  after  the  first  one  or  two  figures  have 
been  found. 

575.   Find  by  Sturm's   TJieorem  the  first  two  figures  of 
each  incommensurable  root  of  tbe  following  equations : 

2.  xz  4-  3.r2  —  32  4-  1  =  o. 

3.  x5  —  5-r4  —  10a;3  +  iox  —  1=0. 

4.  x4  4-  3-T3  —  6x  +2  =  0. 

5.  xs  —  5-r2  +7=0. 

6 .  x5  —  x  -f-  5  =0. 

7.  x7  —  7X6  +  jx5  —  14X*  —  7.T3  +  yx2  +  142;  +1=0. 

8.  x3  —  6x2  +  3.r  +  5  =  o. 

9.  Xs  4-  6a:2  —  t,x .+  9  =  o. 
10.  .r3  +  5a;2  —  ix  4-  2  =  o. 


HORNER'S     METHOD.  267 


HORNER'S     METHOD     OF    APPROXIMATION.* 

576.  Horner's  31ethod  of  finding  the  successive 
figures  of  an  incommensurable  root  is  based  on  the  following 

Problem. — To  find  an  equation  whose  roots  shall  he  less 
than  the  roots  of  a  given  equation  by  a  given  number. 

Solution. — Let  x  be  the  number  by  which  the  roots  of  f(x)  =  o  are 
to  be  diminished,  and  put  y  for  the  unknown  quantity  in  the  trans- 
formed equation.     Then 

y  =  x  —  x'        or       x  =  y  +  x'. 

Substituting  y  +  x'  for  x,  we  have, 

(y  +  x')n  +  Aiiy  +  x')"-1  +  Ai(y  +  x')n~2  .... 

An-ziy  +  x'f  +  An-\  (y+x1)  +  An  =  f(y+x')  =  o.        (i) 

Developing  the  different  powers  of  (y  +  x')  and  collecting  the  terms 
containing  like  powers  of  y,  and  writing  B\,  -Z?2 ,  B3,  etc.,  for  the 
coefficients,  we  have, 

yn  +  By-1  +  Bay"-2 +  B,,-iy  +  BH  =  f(y  +  %')  =  o.        (2) 

Substituting  for  y  its  value,  (x—x'),  this  equation  becomes, 

(x—x)n  +  Bi  (x—x')n~v  +  B2(x—x')11-2 .... 

.&_,  {x-x')  +  Bn  =  f(x)  =  o.        (3) 

If  equation  (3)  be  divided  by  (x—x'),  the  remainder  will  be  Bn  ;  and 
if  the  quotient  be  divided  by  (x—x'),  the  remainder  will  be  Bn—\,  and  so 
on,  by  successive  divisions  by  (x—x'),  the  remainders  will  be  the  succes- 
sive coefficients  of  (2),  beginning  with  the  last.  But  the  first  member  of 
equation  (3)  is  equal  to  f(x) ;  hence  if  these  successive  divisions  by 
(x—x')  be  performed  upon  f(x),  the  remainders  will  be  the  coefficients 
required. 

In  the  following  example  the  transformation  is  effected 
both  by  substitution  and  by  the  method  of  division : 

1.  Transform  the  equation  a8  —  5a*  +  $x—  7  =  o,  to 
another  whose  roots  shall  be  2  less  than  those  of  the  given 
equation. 

1st.  By  Substitution, 
(y  +  2)3  -  5  (y  +  2f  +  3  {y  +  2)  -  7  =  o. 

*  Sed  pasje  307,  Note  6. 


268 


Horner's    method 


Developing  the  equation, 


Or, 


y3  +  by2  +  12 

0  +     8 

=  0. 

—  5I      —  20 

—  20 

+     3 

+    6 

-    7 

y3  +  f  -  sy  -  j 

3  =  0,    ,4ns. 

2d.  By  Division. 

1    -    5     + 

3     -     7     |  2 

+     2    — 

5     — 

5 

3  =  B; 

—     3     - 

3   «-  1 

2     — 

2 

-     1  «-     i 

5  =  52 

2 

J,3     +     yi    _     5^    _     I3     —    Qi         ^4^ 

Notes. — 1.  The  coefficients  of  the  first  terms  of  the  successive  quo- 
tients are  omitted,  as  not  essential  to  the  work. 

2.  The  coefficients  of  the  transformed  equation  are  printed  in  full- 
face  type,  and  preceded  by  an  inverted  comma. 

577.  To  apply  this  by  Horner's  method  of  approximating 
the  incommensurable  roots  of  an  equation,  take  Ex.  1,  Art.  574. 


x3  —  3Z2  +  6x  —  5  =0. 


(0 


"We  found  that  this  equation  has  but  one  real  root,  and  that  this  root 
is  between  1.3  and  1.4. 

To  find  other  figures  of  the  root,  diminish  the  roots  first  by  1,  as 
follows : 


— 

3 

+ 

6     - 

5 

+ 

1 

— 

2     + 

4 

— 

2 

a. 

4  ,- 

1 

+ 

1 

— 

1 

— 

1 

*  + 

3 

+ 

1 

This  gives 


«  +    0 

y*  +  oy!  +  yy  —  1  =  o, 


(2) 


an  equation  whose  roots  are  1  less  than  those  of  the  primitive  equation. 
Its  real  root  therefore  lies  between  .3  and  .4. 


HORNER    S     METHOD. 


269 


Diminishing  its  roots  by  .3,  as  before, 

1     +     0     +     3         —     1 
+    .3     +       .09    +       .927 


3-09 
.18 


.073 


:+     3.27 


and  we  have 


+  .9s2  +  3.27s  —  .073  =  o, 


(3) 


an  equation  whose  roots  are  1.3  less  than  those  of  the  primitive  equation. 
Its  real  root  will  therefore  be  the  remaining  figures  of  the  real  root 
of  the  primitive  equation.  Hence  it  is  less  than  .1,  its  cube  less  than 
.001,  and  its  square  less  than  .01,  and  the  omission  of  the  first  two  terms 
will  make  but  little  difference  in  its  root.  We  may  therefore  find  the 
root  approximately  by  using  only  the  last  two  terms. 


Thus, 

3.27s  —  .073  =  0, 

or 

3.27s  =  .073, 

and 

■073 
z  =  — —  =  .02  + 

3-27 

This  gives,  probably,  the  first  figure  of  the  root  of  (3),  and  the  third 
figure  of  the  root  of  (1). 

Diminishing  the  roots  of  (3)  by  .02,  we  have, 


«  + 


3-27 
.01* 


-  -073 
+  .065768 


.92  +  3.2884  (—    .007232 
,02  +  0.0188 


•94  <  + 
02 


3.3072 


96 


The  new  equation  is 

w3  +  .9610°  +  3.3072W  —  .007232  =  o. 


(4) 


We  know  by  this  result  that  .02  is  not  greater  than  the  first  figure 
of  the  root  of  (3),  for  if  it  were,  it  would  give  a  positive  value  on  being 
substituted  for  x  (Th.  VIII),  but  the  result  of  division  shows  that  it 
gives  a  negative  value.     (Th.  I.) 

In  the  same  manner  that  .02  was  found  from  (3)  we  now  find  the 
next  figure  from  (4)  to  be  .002,  and  proceed  to  diminish  the  roots  of  (4) 
by  .002,  a  process  which  may  be  continued  indefinitely. 


270 


HORNER    S     METHOD. 


578.  The  operation  of  diminishing  the  roots  by  these 
successive  figures  may  be  written  more  compactly  by  omitting 
to  re-write  the  coefficients  of  each  new  equation,  and  using 
them  as  they  stand  when  first  found. 

We  give  Ex.  i  written  in  that  manner,  and  designate  the 
coefficients  of  each  successive  transformed  equation  by  half- 
parentheses  and  full-face  type,  marking  them  by  a  subscript 
figure  to  indicate  whether  they  belong  to  the  first,  second,  or 
third  transformed  equation. 

Thus,  i(  +  3,  indicates  a  coefficient  of  the  first  transformed  equation, 
and  2(3.27  a  coefficient  of  the  second  transformed  equation. 

By  a  careful  examination  the  student  will  see  that  the 
process  is  equivalent  to  the  transformations  above.  Observe 
that  when  a  decimal  is  to  be  added  to  an  integer,  it  is 
annexed  for  greater  economy  of  space. 


i  —  3 

+  i 


+   6 

—    2 


—  5     |  1.3221 

+  4 


—    2 

+     I 

+  4 
—  i 

.(-  1 

.927 

—    I 
+     I 

i(+   3.09 
+     .18 

2(-    .073 
+     .065768 

.(  + 
+ 

0-3 

•3 

+ 

3.27 
.0184 

3(~ 

+ 

.007232 
.006618248 

+ 
+ 

.6 
•3 

+ 

+ 

3-2884 
.0188 

4(~ 

+ 

.000613752 
000331 115 

+ 

.92 

.02 

3(  + 
+ 

3.3072 
.001924 

b(- 

.000282637 

+ 

+ 

•94 
.02 

+ 
+ 

3309*24 

.001928 

3(  + 

+ 

.962 
.002 

4(  + 

+ 

3.311052 
.00009661 

+ 
+ 

.964 
.002 

+ 

3.31114861 

*(  + 

.966i 

Let  the  student  extend  the  above,  and  find  two  or  three 
more  places  of  the  root.  He  will  observe  (hat  it  is  unnecessary 
to  use  more  than  two  places  of  decimals  in  the  first  column, 
four  in  the  second,  and  six  in  the  third. 


HORNER'S     METHOD.  271 

Let  him  also  apply  this  method  to  equations  in  Art.  575, 
of  which  two  figures  of  the  roots  have  already  been  found. 

In  finding  negative  roots,  change  the  signs  of  the  alternate 
terms  beginning  with  x71'1,  and  thus  change  the  signs  of  the 
roots.     (Art.  552.) 

The  positive  roots  of  the  equation  thus  changed  will,  when 
their  signs  are  changed,  be  the  negative  roots  of  the  primitive 
equation. 

2.  Find  the  fifth  root  of  5. 

The  equation  is  x5  —  5  =  o,  which  by  Art.  296,  20,  has  but  one  real 
root,  and,  without  the  application  of  Sturm's  Theorem,  we  know  the  first 
figure  to  be  1.  We  may  therefore  proceed  at  once  to  apply  Horner's 
Method,  as  follows  : 


+ 
+ 

0  + 

1  + 

0 
1 

+ 

+ 

0  + 

1  + 

0  — 

1  + 

5 
1 

+ 

+ 

I    + 
1    + 

1 
2 

+ 
+ 

1     + 
3     + 

4 

4 

+ 
+ 

2       + 
I       + 

3 
3 

+ 
+ 

4i(+ 
6 

5 

+ 
+ 

3     + 

1      + 

6 
4 

(  + 

10 

+ 
+ 

4i(  + 
1 

10 

l(+ 

5 

If  we  now  divide  4  by  5  to  find  the  next  figure,  we  get  .8,  but  this  is 
evidently  too  large  ;  for  by  Art.  556,  the  first  remainder  obtained  in  the 
next  division  must  be  negative.  If,  therefore,  we  use  a  figure  so  large  as 
to  give  a  positive  result,  we  must  reduce  it.  Let  the  student  complete 
the  work  in  this  example. 

3.  Find  the  cube  root  of  1953 125  by  Horner's  method. 

4.  Find  the  real  roots  of  xi  —  x2  -f  2X  —  1  =0. 

5.  Find  one  root  of  x5  —  2.14  —  yc2  —  $x  —  38756  =  o. 

6.  Find  a  root  of  x3  —  2  =0. 

7.  Find  the  roots  of  5.V3  —  %z  —  1  =0. 


373  hoexee's    method. 

579.  To  apply  Sturm's  Theorem  and  Horner's  Method  of 
Approximation,  it  is  not  necessary  to  reduce  the  equation  to 
the  form,  Art.  538.  To  use  synthetic  division  for  the  applica- 
tion of  Horner's  Method,  we  must  however  make  the  first 
coefficient  unity.  This  will  give  for  Example  7,  the  coef- 
ficients 

1  +0  —  .6  —  .2 

8.  Find  the  roots  of  xz  —  jx  +  7  =  o. 

9.  Find  the  roots  of  xs  —  5.?;  —  5=0. 

10.  Find  the  roots  of  x3  +  3a;2  +  3^  +  5  =0. 

580.  The  foregoing  processes  of  finding  the  real  roots  of 
numerical  equations  may  be  summed  np  as  follows: 

1st.  Reduce  the  equation  to  the  form 

xn  4-  A,xn~l  +  A<2xn~2 .  .  .  .  4-  An^x  4-  A„  =  o, 
by  Arts.  539-542. 

2d.  Try  the  factors  of  the  last  term  by  Art.  566  for  com- 
mensurable roots. 

3d.  Divide  out  the  roots  thus  found.     (Art.  544,  Cor.  1.) 

4th.  Apply  Sturm's  Theorem  to  the  depressed  equation,  and 
if  in  the  course  of  the  process  a  common  divisor  between  /(•'') 
and  f  (x)  be  found,  form  two  equations  by  Theorem  XI,  and 
apply  Sturm's  Tlieorem  to  these  equations  to  find  the  situation 
of  the  incommensurable  roots. 

5th.  Find  an  approximation  to  each  of  the  incommensurable 
roots  by  Horner's  Method. 

581.  The  imaginary  roots  may  be  found,  when  there  are 
only  two,  by  removing  the  other  roots  and  reducing  as  a 
quadratic. 


hoknee's    method.  273 

582.  When  fin  equation  has  two  nearly  equal  roots,  it  will 
be  necessary  to  find,  by  Sturm's  Theorem,  a  sufficient  number 
of  figures  to  separate  the  roots. 

For  example,  in  the  following  equation, 

x3  +  ll%2  —  102a;  4-  181  =0, 
we  find,  by  Sturm's  Theorem,  that  the  roots  are  all  real,  two 
being  positive  and  one  negative.     We  also  find  the  positive 
roots  situated  between  3  and  4.     Diminishing  the  roots  by  3, 
gives  the  coefficients  of  the  transformed  equation, 

1  4-  20  —  9  -f  i, 
two  of  whose  roots  are  between  o  and  1. 

Applying  Horner's  Method,  trying  successively  .1,  .2,  etc., 
we  have,  1  +  20     —  9       +1        |  .1 

+      .1  +  2-01  —  '699 
+  20.1  —  6.994+   .  301 
The  remainder  being  -f ,  both  positive  roots  are  either  greater 
or  less  than  .1.     (Theorem  VIII.) 

1  4-  20     —  9       4-1        I  .2 
4-      .24-  4.04  —    .992 
4-  20.2  —  4.96,+    .008 
For  the  same  reason  these  roots  are  both  either  greater  or  less 
than  .2,  and  if  we  were  to  try  .3,  .4,  etc.,  we  should  still  find  a 
positive  remainder.     Horner's  Method,  therefore,  does  not  dis- 
tinguish between  these  roots.     But  by  further  application  of 
Sturm's  Theorem,  we  find  the  positive  roots  of  the   primitive 
equation  are  situated  between  3.2   and  ^.^,  and  by  Horner's 
Method  we  may  then  find  the  remaining  figures. 

Diminishing  the  roots  by  3.2  gives  the  coefficients, 
1  4-  20.6     —  .88       4-  .008       J  .01 
4-       .01  4-  .2061  —  .006739 
4-  20.61  —  .6739,+  -001261 
and  wre  see  that  both  roots  of  the  primitive  equation  are  either 
greater  or  less  than  3.21. 

1  4-  20.6     —  .88       4-  .008       I  .02 

4-       .02  4-  -4124  —  -OQ9352 
4-  20.62  —  .46764—  .001352 
We  now  know  that  one  root  is  greater  and  one  less  than  3.22. 
Let  the  student  find  each  of  these  roots  to  5  decimal  places. 


274  KECUERIKG     EQUATIONS. 


RECURRING     EQUATIONS. 

583.  A  Recurring  liquation  is  one  in  which  the 
coefficients  of  xn~r  and  xr  are  numerically  equal,  the  cor- 
responding coefficients  all  having  either  like  ox  unlike  signs. 

584.  A  Reciprocal  Equation  is  one  whose  roots 
are  reciprocals  of  each  other;  that  is,  if  a  be  a  root,  -  is  also 
a  root. 

Theorem    XIV. 

585.  A  recurring  equation  is  also  Reciprocal. 
Demonstration. — The  general  recurring  equation  is 

x"  +  J-jCC'-i  +  A2xn~2  .  .  .  ±  A2x2  ±  A^x  ±  i  =  o.  (i) 

Substituting  -  for  x, 
x 

—  +  A,  — ,  +  A-  ——i.  .  .  .  ±  A*  —  ±  A,  -  ±  i  =  o.  (2) 

Clearing  of  fractions, 

i  +  Axx  +  A2x-  .  .  .  ±  A.,xn~2  ±  A-lX'1~1  ±  x"  =  o.  (3) 

Which  is    the   same    as   (i) ;    hence    the    equation    is  satisfied   when 

-  is  put  for  x. 
x 

In  equation  (i),  the  double  signs  in  the  second  half  indicate  that 
those  terms  are  either  all  of  the  same  sign  as  the  corresponding  terms  of 
the  first  half,  or  all  of  contrary  sign. 


Theorem    XV. 

586.  A  Recurring  Equation  of  an  odd  degree  has  —  i  or 
+  i  as  a  root,  according  as  the  corresponding  terms  have  like 
or  tinlike  signs. 

Demonstration. — Whenever  the  corresponding  coefficients  have 
unlike  signs,  the  substitution  of  + 1  for  x  will  cause  them  to  cancel  each 


RECURRING     EQUATIONS.  275 

other;  and  when  they  have  like  signs,  —  i  substituted  for  x  will  do 
the  same  ;  since  one  of  these  terms  will  be  an  even  and  the  other 
an  odd  power  of  x. 

Cor. — Such  an  equation  may  have  its  degree  reduced  one 
unit  by  dividing  by  x  —  i    or  x  +  i. 


Theorem    XVI. 

587.  A  Recurring  Equation  of  an  even  {the  2nth)  degree, 
in  which  the  coefficient  of  xn  is  zero  and  the  like  coefficients 
have  unlike  signs,  has  both  -\-i  and  — i  as  roots. 

Demonstration. — Represent  the  equation  by 

x2n  +  J-!*2"-1  +  A2x-><---  ...  —  A2x-  —  A^x  —  i  =  o. 

It  is  evident  that  both  +i  and  —  i  will  cause  corresponding  terms 
to  cancel. 

Cor. — Such  an  equation  may  be  divided  by  x2  —  i,  and 
its  degree  thus  reduced  two  units. 

Theorem    XVII. 

588.  Every  Recurring  Equation  of  an  even  degree,  whose 
corresponding  terms  have  like  signs,  may  be  reduced  to  an 
equation  of  one-half  that  degree. 

Demonstration. — Let  the  equation  be 
x2'1  +  ^41a;2n— i  +  Asx2n~2 .  .  .  +  A„xn  .  .  .  +  A»x2  +  Axx  +  i  =  o.    (i) 
Dividing  by  x"  and  uniting  terms, 

(*  +  L)  +  Al  ^-,  +  _L)  +  As  (;Cn-2  +  _!__) . . .  An  =  o.   (2) 

Make 

then, 


2 

— 

X 

+ 

X  ' 

89 

—  2 

= 

& 

+ 

I 

x-' 

3  _ 

-33 

- 

it'3 

+ 

I 
^3  5 

276  RECUERIXG     EQUATIONS. 

and  in  general   a"»  h may  be  expressed  in  terms  of  2,  the  highest 

power  being  zm. 

Substituting  these  values  in  (2),  we  have  tm  equation  of  the 
nth  degree. 

589.  Eecurring  equations  may  often  be  reduced  as  quad- 
ratics, by  reducing  the  degree  in  accordance  with  the 
preceding  theorems.  The  following  examples  will  make 
the  student  familiar  with  the  principles. 

EXAMPLES. 

1.  x5  —  7>xi  +  2xZ  —  2x2  +  Sx  ~~  l  —  °- ' 

Solution. — By  Theorem  XV,  +  1  is  a  root.  Dividing  out  this  root 
we  have  the  equation, 

xi  —  2X3  —  2X  +  1  =  o. 

Dividing  by  a;2, 

a:2  +  - :  —  2  (x  +  -)  =  o. 
xl        \        ay 


Substituting  s  =  x  +  - , 
x 


z2  —  2  —  2S  =  o ; 

z  =  1  ±  \/3  ; 
x  +  -  =  1  ±  -v/3  ; 

X 

<&  +  1  =  (1  ±  y3)  ■** ; 

=  £AJ5  ±  1  a/±7^.  (a) 


a; 

2 


The  roots  written  separately  are  :     —  1 ; 

I+   ^  +  J  V7^  ; 
2 

LtJ5_i  VT^a; 
2 


2 
1  — 


BINOMIAL     EQUATIONS, 


277 


Note. — Observe,  that  in  equation  (a)  the  double  sign  under  the 
radical  comes  from  the  double  sign  in  the  numerator  of  the  preceding 
term ;  hence,  the  upper  signs  of  these  must  be  taken  together,  and  also 
the  lower.  But  the  double  sign  between  the  two  terms  has  no  depend- 
ence on  the  others,  and  may  therefore  be  taken  either  way. 


X3  —  2X~  +    2X  —   I    =   O. 

x4  —  3a;3  -+-  4X2  —  $X  +  I  =  O. 

x*  +  7X3  —  jx  —  1  =  o. 

3.r5  —  zx4  +  5a;3  —  5.r2  +  2X  —  3  =  0. 

x6  —  3«5  +  5a4  —  5a;2  +  32;  —  1=0. 

x4-  —  x3  +  x2  —  x  +  1  =  o. 

X4  —  X5  +  X  —  I    =:  O. 

ax4  —  2X3  -\-  2x  —  a  =  o. 

5X4  +  Sx3  +  9a;2  +  Sx  +  5  =  o. 


BINOMIAL     EQUATIONS. 

590.  A  Binomial  Equation  is  one  having  but  two 
terms,  and  is  of  the  form 

yn  ±  an  —  o.  (1) 

By  substituting  ax  for  y,  it  takes  the  form 

a"^  ±an  =  o,     or    xn  ±  1  =  o.  (2) 

The  roots  of  (2)  multiplied  by  a  will  give  the  roots  of  (1). 

591.  The  real  roots  of  a  binomial  equation  may  be  found 
by  the  usual  method  of  evolution  or  by  Horner's  method  of 
approximation. 

592.  The  Imaginary  Roots  can  be  found  by  the 
method  (Art.  303)  which  involves  Trigonometry,  or,  if  the 
degree  of  the  equation  be  not  too  high,  by  solving  the 
equation  in  form  (2)  as  a  recurring  equation. 


278 


BINOMIAL     EQUATION'S, 


EXAMPLES. 

i.  Find  the  roots  of  x5  —  i  =  o. 
One  root  i3  i,  which  being  removed  by  division  gives 
x4  +  x3  +  x2  +  x+i  =  o. 


Dividing  bvx2,     x2  +  —  +  x  A 1-1=0. 

x2  x 


Putting 


x  + 


S2  +  2  —  I  =  o ; 


z  =  —  |  ±  yi+I  = 


i  Vs  =  x  +  ~  ' 


&  +  (i^iVl)ai  =  —  ij 
=  -i±iV5±V'(W  ^Y  -  i ; 


!B=i(-I±A/5±  V-IOT2    fJ0. 

The  roots  of  £5  —  i  =  o   are,  therefore,   i, 

i(-i  +  V5  +  V-10-2  V5) 


(-1  +  \/5  -  V-  10-2  fs) 


(—  I  —  y/s  +  V—  IO+2  V$) 


H-I-V5-  V-IO  +  2    ^). 


The  fifth  roots  of  any  other  number  will  be  these  roots  multiplied 
by  the  real  5th  root  of  that  number.  For  example,  the  roots  of 
xb  —  32  =  o   are  the  above,  each  multiplied  by  2. 

Find  the  roots  of  the  following : 

2.  7?  ±1  =  0.  6.     a?  +  7  =  o. 

3.  .r1  ±1=0.  7.     .i-5  —  4  =  0. 

4.  .r6  ±1=0.  8.     a7  -f-  5  r=  o. 


5.     a*  +  5  =  o. 


9.     z-°  —  3  =  o. 


EXPONENTIAL     EQUATIONS.  279 


EXPONENTIAL     EQUATIONS. 

593.  An  Exponential  Equation  is  an  equation  in 
which  one  or  more  of  the  exponents  contains  an  unknown 
quantity. 

i.  Given   ax  =  b  to  find   x. 

Solution. — Taking  the  logarithms  of  both  members, 

x  log  a  =  log  b.     (Art.  460.) 

log  b 
x  =  r-e-. 

log  a 

2.  Given   x®  =  a. 
Solution. — Taking  the  logarithms 

■x  log  x  =  log  a. 

Find   by   inspection   from  the   table   of  logarithms   and   by   trial,  the 
value  of  x. 

For  example,  let  log  a  =  8. 

Then,  x  log  x  =  8, 

and  by  inspection,  x  =  8.6,  nearly. 

3.  Given  10^  =  57  to  find   x. 

4.  Given  27^  =  84  to  find  x. 

5.  Given  x2®  =  13  to  find  x. 

6.  Given  xx+%  =  5  to  find   x. 

7.  In  how  many  years  will  a  sum  of  money  double  at 
compound  interest,  at  6%  ? 

8.  In  how  many  years  will  a  dollars  amount  to  A  at 
compound  interest,  at  v ,  ? 

9.  If  a  young  man  spends  25  cents  a  day  for  cigars,  in  how 
many  years  might  he  buy  a  farm  worth  $5000  with  the  same 
money,  by  depositing  it  once  a  quarter,  at  5%  compound 
interest  ? 


<\ 


r 


w 


APPENDIX. 

PROBABILITIES. 

594.  The  Probability  that  any  event  will  happen  is 
the  ratio  of  the  favorable  to  the  whole  number  of  chances. 
This  is  on  the  supposition  that  the  chances  are  all  equally 
good. 

595.  Let  it  be  known  that  a  bag  contains  10  balls,  num- 
bered from  i  to  10.  If  one  ball  be  drawn,  and  there  be  no 
reason  why  it  should  be  one  number  rather  than  another,  we 
say  there  is  one  chance  in  ten  that  it  will  be  a  particular 
number,  as  No.  2  or  No.  5.  The  chance  then  that  No.  5  will 
be  drawn  is  ^,  there  being  10  events  each  equally  likely  to 
happen  and  only  one  of  them  being  favorable. 

If,  of  the  10  balls,  3  are  white,  2  black  and  5  red,  the 
chance  that  a  red  ball  will  first  be  drawn  is  ■£$  or  \  ;  that  a 
black  ball  will  be  drawn  is  ^  or  \ ;  and  that  a  white  ball 
will  be  drawn  is  ^. 

The  chance  that  the  first  ball  drawn  will  be  either  black 
or  red,  is  ^  +  \  =  -^ ;  and  the  chance  that  it  will  be  either 
white,  black  or  red,  is  \  +  \\--r$  =  1.  In  this  case  the  chance 
becomes  a  certainty,  and  is  represented  by  1. 

596.  If  another  bag  contain  4  white,  3  black  and  3  red 
balls,  the  chance  of  drawing  the  first  time  a  red  ball  from  this 
is  T%,  while  the  chance  of  drawing  a  red  from  the  first  bag 
is  t%.  The  chance  that  both  balls  will  be  red  is  the  product 
of  the  chances  for  each,  or  T5„  x  ^  =  -^.  This  is  evident, 
for  any  one  of  the  balls  in  the  first  bag  may  be  drawn  wit li 
any  ope  in  the  other,  making  the  whole  number  of  chances 


PROBABILITIES.  281 

the  product  of  the  whole  number  in  one  bag  by  the  whole 
number  in  the  other.  Also,  since  each  of  the  favorable 
chances  in  one  case  may  happen  with  any  one  of  the  chances 
in  the  other,  the  product  of  the  favorable  chances  in  one  case, 
multiplied  by  the  favorable  chances  in  the  other  case,  will  give 
the  number  of  favorable  chances  when  the  two  events  are  to 
occur  together. 

EXAMPLES. 

i.  What  is  the  chance  of  throwing  sixes,  with  two  dice,  at 
the  first  throw  ? 

Solution. — The  chance  that  either  of  the  dice  will  turn  up  a  six 
is  evidently  J.     Hence  the  chances  of  double  sixes  is   a  x  i  =  ^\,  Am. 

2.  What  is  the  chance  of  throwing  sixes  twice  in 
succession  ? 

Solution. — The  chance  that  two  throws  will   give  the  same  is 

Sf    X     36    -    T5S6>   A.IIS. 

3.  What  is  the  chance  that  a  throw  of  two  dice  will  be 
greater  than  6  ? 

4.  What  is  the  chance  that  in  drawing  100  numbers  from 
a  box,  any  three  numbers  will  be  drawn  consecutively  ? 

5.  What  is  the  chance  that  three  points  taken  at  random 
on  a  circle  will  be  on  the  same  semicircle  ? 

6.  What  is  the  chance  that  two  coppers  tossed  at  random 
will  both  turn  up  heads?  What  is  the  chance  that  three 
coppers  will  do  the  same  ? 

7.  What  is  the  chance  that  six  coppers  will  turn  up  half 
heads  and  half  tails  ? 

8.  What  is  the  chance  of  drawing  three  white  balls  suc- 
cessively from  a  bag  containing  10  white,  7  black  and  5  red 
balls,  the  ball  drawn  being  replaced  before  the  next  draw  ? 

9.  What  would  the  chance  be  in  Prob.  8,  if  the  ball  be  not 
replaced  ? 

10.  What  is  the  chance  that  one  of  each  color  will  be 
drawn  in  the  first  three  draws,  not  replacing  balls  drawn  ? 


282  cardan's    formula. 


CARDAN'S     FORMULA. 

597.    Cardan's  formula  for  the  reduction  of  cubic  equa- 
tions is  obtained  as  follows : 

The  general  form  of  a  cubic  equation  is 

a?  +  ax2  +  bx  -\-  c  =  o:  (i) 

From  this  equation  the  second  term  may  be  made  to  disap- 
pear by  the  substitution  of  y for  x,  giving 

(y--)  + a  \y  - 1)  + h  \y  -  -)  + c  =  °- 

This  diminishes  each  root  of  the  equation  by  the  quantity 
-^     (Art.  S76.) 

Developing  and  reducing, 

=  o. 


if  +  of  —  \a% 
+  I 


y  +  -fcc? 

-lab 


+  c 
Substituting^  for  b  —  ^a2,  and  q  for  faa3  —  \ab  +  c  gives 

f  +  py  +  q  =  o.  (2) 

Making  y  =  z  —  ^ 

p3 
the  equation  becomes  z?  +  qzs— —  -  =  o, 

27 

which  by  Art.  329,  (A),  gives 


=  -  9 '  ±  \/r 


2  -n3 

4         27 


cardan's    formula.  283 


and  zt=(-£±Vq-  +  ^J,    therefore, 


V      2  4       27/ 

or,  rationalizing  the  denominator  and  omitting   the  double 
signs,  because  they  give  no  more  values  than  single  signs, 


598.  In  equation  (2),  the  coefficient  of  y2  being  0  the  sum 
of  the  roots  is  o  (Art.  547,  Cor.  3);  hence,  representing  two 
of  the  roots  by  m  ±  VJi,  the  third  root  will  be  —  2m.  These 
roots  give  the  equation 

Vz  —  (3m?  +  n)  V  +  2  (m5  —  mn)  —  °-  (Art.  547,  Cor.  2.) 
Equating  with  (2)  gives  the  identical  equation, 

ys  +  py  +  q  =  y3  —  (3m2  +  n)  y  +  2  (wi3  —  w*w)  > 

p  =  —  (3™2  +  w)  ; 
q  =  2  (m3  —  ?ww). 


Hence,        |/ ^  +  ^  =  (^2  -  fw)  V-  3».  (3) 

The  roots   —  2m   and  m  ±  Vw  admit  of  three  cases  : 

1.  When  n  is  positive  and  the  roots  are  all  real. 

2.  When  n  is  o  and  tivo  of  the  roots  are  equal. 

3.  When  n  is  negative  and  tivo  of  the  roots  are  imaginary. 

The  first  case  makes  the  second  member  of  (3)  imaginary, 
and  therefore  the  terms  of  formula  (A)  become  imaginary. 

The  second  and  third  cases  make  (3)  real,  and  formula 
(A)  real.    Hence, 


284  cardan's    formula. 

When   the   roots   of  a  cubic  equation   are   all  real  and 
unequal,  Cardan's  formula  has  its  terms  imaginary. 

This  will  occur  when  p  [Eq.  (2)]  is  negative  and 

-  <  —  — • 

4  27 

599.   The  following  examples  illustrate  the  application  of 
the  formula  to  each  of  these  cases: 

1.  Find  the  roots  of  z3  +  6x  —  20  =  o. 

By  the  formula, 

< 

X  =  (10  +  •\/ioo  +  8)4  +  (10  —  /y/100  +  8)s 

=  (10  +  6  ^3)*  +  (10  -  6  -\/3)* 

=  1  f  ^3  +  1  -  \/3 
=  2. 

Removing  this  root  from  the  equation, 

1  +  0  +  6  —  20  \  2 

2  +  4  +  20 

2  +  10 

The  depressed  equation  is 

x*  +  2X  +  10*=  o  ; 

and  x  —  —  1  ±  y/—  9  =  —  1  ±  3  \/^. 

The  roots  are  2,        —  1  —  i  3,        and        —1—53. 

2.  Find  the  roots  of  a-3  —  3%  —  2  =  o. 

By  the  formula, 

x  =  (1  +  \/i  —  i)1  +  (1  —  V1  —  J)* 
=  1  +  1  =  2. 

Dividing  by  x  —  2  and  reducing  the  depressed  equation,  we  find 

X  =  —  I  ±  0. 


CONTINUED     FEACTIONS.  289 

fraction,  |,  is  too  great ;  and,  consequently,  2^  being  greater 
than  the  true  denominator,  the  fraction, 

1  x        3 


r,      _l_      I  7  n    ' 

2   +    3"  ^  7 

will  be  /ess  than  the  true  value  of  the  continued  fraction. 

604.  Similar  reasoning  will,  evidently,  hold  in  respect  to 
any  number  of  terms,  and  will  apply  equally  to  the  general 
form  (2),  as  to  the  particular  example  we  have  considered. 

Hence, 

If  we  include  in  the  reduction  an  odd  number  of  partial 
fractions,  the  result  ivill  be  too  great;  if  an  even  number, 
the  result  will  be  too  small. 


605.   The  fractions, 
1  1 


a{  x'  1' 

<h  +  -  «i  +  - 

a2  1 

a9  +  ~ 
a. 


etc., 


are  approximate  values  of  the  given  fraction,  and  are  some- 
times called  approximating  or  converging  fractions,  or  simply 
Convergents. 

It  is  evident  that  the  true  value  of  the  continued  fraction, 
lying  between  two  successive  approximate  values,  differs  from 
either  of  them  less  than  they  differ  from  each  other. 


606.   We  have 

1         1 

a,  —  «j ' 
1                       a2 

1st  approx.  value 

2d        "           " 

a.2a3  +  1 

1 

1         a,a2  -f  1' 
Ui  -\ 

«2 

I 

«2    +    - 
«3 

1 

<h  +  - 

1 
a.2  +  - 
a3 

a 

\          a-J 

+  i)(<3  +  ai' 

....  3d  approximate  value. 
13 


290  CONTINUED     FRACTIONS. 


We  shall  evidently  find  the  fourth  approximate  value,  or 
verger 

Thus, 


convergent,  by  substituting,  in  the  third,  aa  +       for  a 

a4 


(a,a3  +  i)a4  +  Oj 


[(«,«,  +  r)  a3  +  fli]  a4  +  axa,  +  i 

is  the  fourth  convergent. 

We  obviously  find  the  numerator  and  denominator  of  the 
third  convergent  by  multiplying  those  of  the  second  by  the  third 
partial  denominator,  and  adding  those  of  the  first  convergent. 

We  find,  in  like  manner,  the  fourth  convergent  from  the 
terms  of  the  second  and  third. 

607.  To  show  the  generality  of  this  law,  let  it  be  admitted 
to  hold  good  as  far  as  the  nth  convergent  (i.  e.,  the  convergent 
corresponding  to  a„). 

T  ,     .      L     M    N         ,  P  ,      . .  , 

Let  also  -=7,  -jp,  -==>,  and  -=  be  the  convergents  cor- 
responding to  cf„_o,  «„_] ,  a„,  and  an+l. 

Then,  since  the  nth  convergent  is  formed  according  to  the 
above  law,  we  shall  have 

N_        Man  +  L  . 

JST  ~  M'on  +  L1'  {3) 

If  now  we  substitute  in  -==. ,  a„  A for  a„ ,  we    shall 

-ft'  '  «n+i 

P 

obviously  find  -=-,•     Thus, 

P_        M\a"  +  T~)  +  L  {Man  +  L)an+,  +  M  . 

P'  ~  \rL    ,     '   \   ,    t.  ~  (M'a»  +  L')a^  +  M' ' 


M'^+i-)+L' 


or,  from  (3),  -^  =  ^      +  j/'"  (4) 

Consequently,  if  the  law  holds  good  for  //  convergents,  it 
will  for  n  +  1.     Hence, 


CONTINUED     FRACTIONS.  291 

608.  To  Find  the  Numerator  and  Denominator  of  any  Con- 
vergent after  the  Second,  as  the  (n  +  i)th,  we  have  the  following 

Eule. — Multiply  the  numerator  and  denominator  of  the 
nth  convergent  by  the  (n  +  i)th  partial  denominator,  and  add 
to  the  products,  respectively,  the  numerator  and  denominator 
of  the  (n  —  i)th  convergent. 

609.  The  numerator  and  denominator  of  any  convergent 
must  be  respectively  greater  than  those  of  the  preceding ;  each 
numerator  and  each  denominator  being  at  least  equal  to  the 
sum  of  the  two  next  preceding. 

610.  Moreover,  each  convergent  is  found  by  substituting 
in  the  preceding,  for  the  last  partial  denominator,  an  expres- 
sion known  to  approach  more  nearly  to  the  true  denominator. 

Hence,  evidently,  each  convergent  approximates  more 
closely  than  the  preceding  to  the  true  value  of  the  continued 
fraction. 

i.  Find  the  successive  convergents  of  the  continued 
fraction, 

i 


2  + 


I 


I 


I 

2  +- 


Am*       I       I      3       4       an(\    3SI 
■a-llb.     •$,    ■$,   -g,   yj,    dim    g^y. 

The  first  four  convergents  are  approximate  values  of  the 
continued  fraction;  the  last,  |f|,  is  the  true  value. 

611.   A  continued  fraction  is  sometimes  mixed,  or  made 
up  of  a  whole  number  and  a  fraction.     Thus, 

i 
3  + 


i 


2  + (5) 

3  +  sTetc. 


292  CONTINUED     FRACTIONS. 

Iii  such  cases,  the  integral  part  may  be  reserved  and  added 
to  the  convergents;  or  it  may  be  taken,  with  i  as  a  denomina- 
tor, for  the  first  convergent. 

612.  Thus,  in  the  above  example,  we  shall  have  the  con- 
vergents, 

,1      ,3       ,16  nr        3       7       2  4       127 


This  form,  "  "*"  i  (6) 


is  sometimes  assumed  as  the  general  form  of  a  continued  frac- 
tion; the  place  of  the  integral  part,  when  it  is  wanting,  beiug 
filled  with  o. 

In  that  case,  the  first  convergent  is  evidently  too  small, 
the  second  too  great,  and  so  on,  those  of  an  odd  order  being 
too  small,  and  those  of  an  even  order  too  great.     (Art.  604.) 

Note. — If  the  integral  part  be  zero,  the  first  convergent  will  of  course 
be  £. 

613.  If  the  second  convergent  of  Art.  606  be  subtracted 
from  the  first,  the  remainder  is  unity  divided  by  the  product 
of  the  denominators.  If  the  third  be  subtracted  from  the 
second,  the  remainder  is  minus  unity  divided  by  the  product 
of  the  denominators. 

Suppose  it  has  been  proved  that  this  law  extends  to  n  —  1 
convergents ;  that  is, 

r  t  nr>         T'  nr  1     , 

(7) 


L       M        LM'  -  L'M 

L'      M'  ~        L'M' 

±  1 
~  L'M' ' 

M 
M' 

N        M        Man  +  L 
N'  ~  M'       M'an  +  L' 

L'M—  LM' 

LM'  -  L'M 

Then    -^7  —  -™  =  -m  — 


M'N'  M'N' 


(8) 


the  numerator  of  which  is  the  same  as  that  of  (7),  with  a 
contrary  sign.  Hence,  the  principle  proved  in  regard  to  the 
first  three  convergents,  applies  equally  to  the  whole  series.    For, 


CONTINUED     FRACTIONS.  293 

If  each  convergent  he  subtracted  from  that  which  next  pre- 
cedes, the  numerator  of  the  difference  will  he  ±  i,  and  the 
denominator  will  he  the  product  of  the  denominators  of  the  two 
convergent^. 

614.  Again,  the  true  value  of  the  continued  fraction  lies 
between  any  two  successive  convergents,  and  differs  from  either 
of  them  less  than  they  differ  from  each  other.     (Art.  605.) 

M 
That  is,  the  convergent  -^7,  differs  from  the  true  value  of 

the  continued  fraction  by  less  than 


M'N' 
But  (Art.  608),  M'  <  N, 

and        .-.        M't  <  M'N'. 

•'•   mW  <  W*    Hence' 

Cor.  1. — The  error  in  taking  any  convergent  whatever  for 
the  true  value  of  the  continued  fraction  is  numerically  less  than 
unity  divided  by  the  square  of  the  denominator  of  that  con- 
vergent. 

615.  The  denominator  of  each  convergent  is  greater  than 
the  next  preceding  by  some  whole  number.     (Art.  609.) 

Hence,  if  the  fraction  be  infinite,  we  may  find  a  convergent 
whose  denominator  shall  be  greater  than  any  given  quantity; 
and,  consequently, 

Cor.  2.  —  We  may  find  a  convergent  which  shall  differ  from 
the  true  value  of  the  continued  fraction  by  less  than  any  given 
quantity. 

616.  Suppose  that  M  and  M'  have  a  common  divisor,  D. 
Then  D  will  of  course  divide  I'M  and  LM',  multiples  of 

M  and  M' ,  and  consequently  the  difference  of  those  multiples, 
LM'  -  LM  =  ±  1. 


294  CONTINUED     FRACTIONS. 

Therefore  D  must  divide  ±  \,  which  has  no  integral 
divisor  but  unity. 

.*.    D  =  i.    Hence, 

Cor.  3. — Every  convergent  is  in  its  lowest  terms. 

617.  One  of  the  most  obvious  uses  of  continued  fractions 
is  to  express  approximately,  in  small  numbers,  fractious  whose 
terms  are  large.     Thus, 

17         11  1  1 


3  +  tt        3  +  rr        3  + 


17        "17  17  „    ,    1 

-5-  2  +  o' 


Here  we  first  divide  both  numerator  and  denominator  of 
4,£  by  17.  We  then  reduce  -ff  to  a  mixed  number,  3-^,  and 
again  divide  both  terms  of  -fa  by  8  and  reduce  to  a  mixed 
number,  and  so  on. 

These  operations  evidently  produce  no  change  in  the  value 
of  the  given  fraction. 

Kow  the  several  convergents  of  the  continued  fraction 
found  are  ^,  f,  and  ££. 

I  I Q— 

We  find  -  =  — - ,  too  great : 

3         59  J 

and         -  =  — -,  too  small, 
7         59 

but  differing  from  the  true  value  by  ouly  7fj. 

2.  If  the  fraction  proposed  had  been  -ff,  we  should  have 
found 

<q  8  1  1 

?r  =  3  +  —  =  3  +  -  =  3  + "J 

17  17  17  1 

8  +8 

and  the  convergents,  3,  -J,  and  £f.     (Art.  611.) 

618.  This  reduction  of  a  common  to  a  continued  fraction 
is  evidently  effected  by  applying  to  the  terms  of  the  given 
fraction  the  process  of  finding  the  greatest  common  divisor,  the 
several  quotients  forming  the  successive  partial  denominators. 


CONTINUED     FRACTIONS.  205 

619.  If  it  be  required  to  transform  any  quantity  whatever, 
as  x,  into  a  continued  fraction,  the  nature  of  continued 
fractions  will  sufficiently  indicate  the  following 

Kule. — I.  Find  the  greatest  integer  contained  in  x,  and 
denote  it  by  a  ;  and  denote  the  fractional  excess  of  x  above  a  by 

—  •     Then    x  =  a  -\ .*.  Xi  =  -      -  >  i. 

xx  xx  x  —  a 

II.  Find  the  greatest  integer  contained  in  xx ,  and  denote  it 
by  ax,  and  denote  the  fractional  excess  of  xx  above  axby  —  •     Then 

x,  =  a,  +  -• 

Xi 

III.  Apply  the  same  process  to  x2,  and  so  on. 
Thus, 

1  l  .    l 

x  =  a  -\ —  =  a  -\ =  a  -\ 

xx  i  ,  •  i 

«i  +  -  «i  + 


x*  i        , 

a*  -\ —  >  etc. 
xs 

If  x  <  i,  we  shall  have  a  =  o. 

We  shall  always  have  xx,  xs,  etc.,  >  i. 

For  if  xx  =  or  <  i,  we  have  -  =  or  >  i,  and  a  is  not 

xx 

the  greatest  integer  contained  in  x. 

620.  Whenever  we  find  a  denominator.  xn,  equal  to  a 
whole  number,  we  shall  have  xn  =  an)  and  the  continued 
fraction  will  terminate. 

This  will  happen  if  the  quantity  x  can  be  exactly  expressed 
by  a  common  fraction. 

If  the  quantity  is  not  equal  to  a  common  fraction  (i.  e.,  if 
it  is  incommensurable),  the  continued  fraction  will  extend  to 
infinity. 


296  CONTINUED     FRACTIONS. 

621.  i.  Given  n  =  3. 141 59,  employing  only  five  decimal 
places.  (Art.  42,  4th.)  Eeduce  n  to  a  continued  fraction, 
and  find  approximate  values. 

Ans.    n  =  3  _j 

7  +  - 


1 

i5  + 


H ,  etc. 

25 
Convergents,  3,  3.2-,  £§£,  fff,  etc. 

Note. — The  second  approximate  value,  "73,  was  found  by  Ar- 
chimedes ;  the  fourth,  fff,  by  Adrian  Metius. 

2.  The  common  or  tropical  year  consists  of  365.242241 
mean  solar  days.     Find  approximate  values  for  this  time. 

Ans.    365 1,  3652V,  365/3,  365T6T'  etc 

Note. — The  third  approximation  shows  an  excess  of  the  solar  year 
above  365  days  of  ^  of  a  day.  To  preserve  the  coincidence  between  the 
solar  and  civil  year,  therefore,  eight  years  in  thirty-three  must  contain 
366  days  each.  That  is,  a  day  must  be  added  to  every  fourth  year  seven 
times  in  succession,  and  the  eighth  time  to  the  fifth  year. 

3.  The  sidereal  month  (i.  e.,  the  time  of  the  moon's  sidereal 
revolution)  consists  of  27.321661  days,  or  the  moon  revolves 
1000000  times  in  2732 1661  days.  Find  approximate  values  of 
this  ratio.  Ans.    27,  -8/,  ?££-,  t^V-j  ctc. 

Note. — These  ratios  show  that  the  moon  revolves  about  3  times 
in  82  days  ;  28  times  in  765  days ;  or,  more  exactly,  143  times  in  3907  days. 

622.  Continued  fractions  are  also  employed  in  finding  the 
roots  of  equations,  and  in  extracting  the  roots  of  numbers. 

1.  Extract  the  square  root  of  3;  i.e.,  find  a  root  of  the 
equation, 

&  —  3  =  °-  0) 

Here  a;=H 

Diminishing  the  roots  of  (1)  by  1,  we  have, 

f.  4.  2lJ  _  2  =  o,  (2) 

an  equation  whose  roots  are  equal  to  -• 


CONTINUED     FRACTIONS.  297 

Transforming  (2),  we  find, 

2.r,2  —  2.r,  —  1  =  0.  (3) 


This  gives,  xt  =  1  H 

Transforming  (3)  in  the  same  manner  as  (1),  we  have, 

X?  —  20i\  —  2=0,  (4) 

and  x2  =  2  -) 

x3 

"We  find,  in  like  manner, 

2X<?  —  2X3  —  1   =0,  (5) 

which  being  the  same  as  (3),  will  have  the  same  roots,  and  will 
give  rise  to  transformed  equations  like  (4)  and  (5). 

Hence,  we  shall  have  a  repetition  of  the  equations  (3)  and 
(4),  and  of  their  roots,  of  which  1  and  2  are  the  integral  parts, 
in  endless  succession. 

.*.    x  =  1  H =  1.732,  etc. 

2  + 


1  H    - ,  etc. 
2 


The  convergents  are  1,  2,  j,  |,  ff>  |f,  |i    f£. 

623.  A  continued  fraction  of  this  kind,  in  which  any 
number  of  the  partial  denominators  are  continually  repeated 
in  the  same  order,  is  called  Periodic. 

624.  It  will  be  found  that  every  incommensurable  root  of 
an  equation  of  the  second  degree  may  be  expressed  by  a 
periodic  continued  fraction. 

Of  course,  when  the  first  period  is  found,  such  a  fraction 
may  be  developed  to  any  extent  by  simply  repeating  the 
period. 

2.  Extract  the  square  root  of  2. 

Convergents,  1,  f,  |,  \\,  f£,  f$,  etc. 


MISCELLANEOUS    EXAMPLES   AND 
PROBLEMS. 


i.  Eeduce  the  following  fraction  to  its  lowest  terms, 
Xs  —  5a;2  —  4X  +  20 
mmmmm  x3  +  5a;2  —  42;  —  20 

2.  Add  -2 to  — -L_^_. 

5«  +  3*        7  ft  -h  9X 

0  ,  .  a  —  a;  2«+z 

3.  bubtract  — 5 =  from  — —  • 

2  a2  +  3aa;  +  a:2  «2  —  a;2 

_   ,         4X  +  1        5a;  —  1 

4.  Keduce =  x  —  2. 

'5  3 

5.  A  travels  5  miles  an  hour,  and  B  starts  on  the  same 
road  3  hours  later  than  A  and  travels  5 \  miles  an  hour. 
When  will  B  overtake  A  ? 

6.  Find  the  time  between  5  and  6  o'clock  when  the  hour 
and  minute  hand  of  a  watch  are  together. 

7.  Find  the  square  root  of 

4a;2  —  i2xy  +  9?/2  +  42:2:  —  6yz  +  z2. 

8.  Find  the  greatest  common  divisor  of 

:<?  —  4,     x2  +  1  ox*  -f  16,     and     x%  —  jx  —  18. 

9.  Find  the  square  root  of 

a6  —  4a5  +  6a4  —  8a3  +  9a2  —  4a  +  4. 

10.  Eeduce  the  equation 

(x  —  3)3  —  3  fa  —  2)3  +  3  (a:  +  i)3  —  x5  +  a;  —  9  =  o. 

11.  Find  how  much  water  must  be  mixed  with  80  gallons 
of  spirit,  which  cost  5  dollars  a  gallon,  so  that  by  selling 
the  mixture  at  4  dollars  a  gallon  there  may  be  a  gain 
of   10;;. 


MISCELLANEOUS     EXAMPLES.  299 

12.  A  person  walks  7^  miles  in  2  hours  17 \  minutes,  and 
returns  in  2  hours  20  minutes.  His  rates  of  walking  up-hill, 
down-hill,  and  on  level  road  were  3,  $h  an(l  2>\  miles 
per  hour,  respectively.  Find  the  distance  travelled  on 
level  road. 

13.  A  man  bought  a  house  which  cost  him  4%  on  the 
purchase  money  to  put  it  in  repair.  At  the  end  of  one  year, 
having  received  no  rent,  he  sold  it  for  $1192,  by  which  he 
gained  10%  on  the  original  cost,  besides  paying  him  5^  on 
his  investment  as  interest  for  the  year.  What  did  he  pay  for 
the  house  ? 

14.  In  a  town  meeting  a  resolution  was  adopted  by  a 
majority  equal  to  ^  of  the  number  voting  with  the  minority; 
but  if  100  of  those  voting  with  the  majority  had  voted 
with  the  minority,  the  majority  in  favor  of  the  resolution 
would  have  been  only  1.  Find  the  number  of  voters  on 
each  side. 

15.  Reduce  Vx  —  Va  +  yx  -fa  —  b  =  Vb. 

16.  Reduce  [(x  —  a)2  +  tab  +  b2]*  =  x  —  a  +  b. 


„   .,  x  —  Vx2  —  a2  (x2  -f  ax)?  —  (x%  —  a%Y 

17.  Reduce 


(x  +  Vx2  —  a2)*  (x2  —  a2)~i 


18.  Reduce  2x  Vi  —  x4  =  a  (1  +  xf). 

19.  Vx  —  or1  —  V*  —  x  l  =  (x  —  1)  or1. 

20.  A  and  B  start  together  to  walk  around  a  circular 
course.  In  half  an  hour  A  has  walked  3  complete  circuits 
and  B  4^.  Assuming  that  each  walks  at  uniform  speed,  tind 
when  B  next  overtakes  A. 

21.  The  distance  from  A  to  B  is  15  miles.  The  road  is 
up-hill  for  the  first  5  miles,  then  level  for  4  miles,  and  then 
down-hill  the  rest  of  the  distance.  A  man  walks  from  A 
to  B  in  3  hours  52  minutes,  and  back  in  4  hours;  he  then 
walks  half  way  to  B  and  back  in  3  hours  55  minutes. 
Find  his  rate  of  walking  up-hill.  down-hill7  and  on  level 
ground. 


300  MISCELLANEOUS     EXAMPLES. 

X2  b2 

22.  If    11  varies  as    ,   and    if   when   x  =  -  ,   y  = 

J  a  +  x  a      * 

3 


a6 
j — Y%,  find  the  equations  between  x  and  y. 


a1  + 

23.  Find  the  sura  of  9   terms  of  an  equidifferent  series 
whose  middle  term  is  18. 


24.  Find  the  sum  of  n  terms  of  the  series, 
1  1  1 


1  +  V2       3  +  2  V2       7  +  5  V2 


etc. 


25.  A  number  consists  of  3  digits.  The  whole  number  is 
equal  to  the  square  of  the  number  formed  by  the  first  two 
digits;  also  the  first  digit  exceeds  twice  the  second  by  unity. 
"What  is  the  number  ? 

26.  Prove  that  the  number  of  ways  in  which  m  positive 
signs  and  n  negative  signs  may  be  placed  in  a  row  so 
that  no  two  negative  signs  shall  be  together,  is  equal 
to  C  „  . 

27.  A  and  B  start  at  the  same  time  and  travel  towards  each 
other.  In  7  days  A  is  5  miles  more  than  his  own  day's  jour- 
ney nearer  the  half-way  house  than  B.  In  10  days  both  have 
passed  the  half-way  house  and  they  are  100  miles  apart,  and 
B  is  3  days  longer  than  A  upon  the  whole  journey.  Required 
their  distance  apart  at  starting  and  rate  of  walking. 

28.  A  farmer  sowed  one  bushel  of  wheat,  and  the  second 
year  sowed  all  the  first  year's  crop,  and  thus  continued  sowing 
each  year  the  whole  crop  of  the  preceding  year.  The  10th 
year  the  product  was  1048576  bushels.  What  was  the  yearly 
rate  of  increase,  on  the  supposition  that  it  was  the  same, 
each  year  ? 

29.  Two  men  start  from  different  points,  at  the  same  time, 
to  walk  towards  each  other;  when  they  meet,  one  of  them 
turns  back,  and  on  reaching  his  starting-point,  again  turns 
and  walks  .in  the  same  direction  as  at  first.  Each  arrives  at 
the  other's  starting-point  at  the  same  time.  Where  did  they 
first  meet?  What  is  the  ratio  of  their  rates  of  walking? 
Where  did  they  meet  the  second  time? 


MISCELLANEOUS     EXAMPLES.  301 

30.  A  man  wishes  to  surround  a  given  area  by  hurdles. 
Placing  them  one  foot  apart,  he  lacks  80 ;  and  putting  them  a 
yard  apart,  he  has  50  hurdles  too  many.  How  many  hurdles 
has  he,  and  at  what  distance  apart  must  they  be,  so  as  just  to 
enclose  the  space  ? 

31.  In  the  bottom  of  a  cistern  containing  192  gallons  of 
water,  two  outlets  are  opened.  After  3  hours,  one  of  them  is 
stopped,  and  the  cistern  is  emptied  by  the  other  in  1 1  hours. 
Had  6  hours  elapsed  before  the  stoppage,  it  would  have  re- 
quired only  6  hours  more  to  empty  it.  How  many  gallons 
did  each  outlet  discharge  in  an  hour,  supposing  the  discharge 
uniform  ? 

32.  Seduce  (4  +  $x  —  x~)?  =  2*2$  +  (x2  +  32;  —  4)^. 

33.  Find  the  relation  between  the  coefficients  of  ax2  -f  bx 
-f-  c  =  o,   that  one  root  may  be  one-half  the  other. 

34.  Divide  1 1 1  into  three  parts,  such  that  the  products  of 
the  parts  taken  two  and  two  may  be  in  the  ratio  of  4, 5,  and  6. 

35.  Show  that  the  number  of  ways  in  which  mn  things 

can  be  divided  among  m  persons  so  that  each  shall  have  n  of 

\mn 
them,  is  >  \Z^- 

36.  The  mth  term  of  an  equidifferent  series  is  - ,  and  the 

71th  term  is  —  ■     Show  that  the  sum  of  mn  terms  is  -       —  ■ 
m  2 

37.  Find  the  sum  of  n  terms  of  the  reciprocals  of  an  equi- 
multiple series  whose  first  term  is  a  and  the  multiplier  m. 

38.  Find  #„  of  2$,  4^,  81V    1 6 A    etc. 

39.  If  a  is  an  equidifferent  mean  between  b  and  c.  and  c 
an  harmonic  mean  between  a  and  b,  show  that  b  is  an  equi- 
multiple mean  between  a  and  c. 

40.  If  n  is  a  positive  integer  and  x  a  positive  fraction  less. 

j  xnJr^-       1  xn 

than  1,  show  that —  < 

n  4-  1  n 

41.  If  a  and  b  are  positive,  and  m  a  positive  fraction  less 
than  1,  show  that  (a  -f  b)m  a>~m  <  a  +  mb. 


302  MISCELLANEOUS     EXAMPLES. 

„.         \  x2  4-  y  =    7  )   to   find   all  the  values  of  x 

42.  Given  {  ,  -  , 

{  x  +  yl  =  n  \  and  y. 

43.  Reduce 

ax  =  by  —  y2 ; 
a;2  =  y2  +  (/;  —  ?/)2. 

44.  Prove  that  w5  —  n  is  always  divisible  by  30 ;  and,  if  n 
be  odd,  by  240. 

45.  The  income  of  a  certain  railroad  company  would  pay  a 
dividend  of  dc/0  if  there  were  no  preferred  stock  ;  but  8400000 
is  such  stock,  and  is  guaranteed  l\%,  the  ordinary  stockholders 
receiving  only  5$.     Find  the  amount  of  ordinary  stock. 

46.  The  population  of  a  certain  town  in  1820  was  2375; 
in  1830,  2948;  in  1840,  3800;  in  1850,  5005;  in  1S60,  6636; 
and  in  1870,  8768.  By  the  same  law  of  increase,  find  the 
population  in  1845,  I^54,  1862,  and  1880. 

47.  A  man  has  a  plank  whose  ends  are  of  unequal  width. 
Find  the  distance  from  the  narrow  end  that  it  must  be  cut,  to 
make  the  parts  equal. 

48.  Find  the  scales  of  the  following  series: 

1,     4X,     i8x2,     So.c3,     35  6.?4,     etc. 
1,    2x,    3.T2,    8X3,     13X4,    30.T5,    5  52*,    etc. 

49.  The  population  of  a  country  increases  25 %  every 
10  years.     In  what  time  will  it  double? 

50.  If  the  student  who  is  attempting  to  solve  this  problem 
belongs  to  a  class  of  50,  of  whom  -^  cannot  solve  it  and  T4„  can 
solve  it,  and  of  the  remainder  f  stand  2  chances  to  1  to  solve 
it.  and  f  stand  an  even  chance  to  fail,  what  is  the  chance  that 
he  will  be  successful  ? 


FORMULAS. 


9- 
10. 

ii. 

12. 

14. 

16. 

17- 
18. 


19. 


20. 


(a  +  x)  (a  —  x)  =  a2  —  x2. 

(a  4-  x)2  =  a2  +  2  ax  4-  ^2. 


(a  -  z)2 


2  ax  +  aA 


fl  ( 7)    J  \ 

(a  +  z)re  =  an  +  Ha*-1 2:  4 ^ *  a^z 

2 


2  r2 


,   w  (n  —  1 )  O  —  2 ) 
4 ~ ■  an~sx3  +  eta 


If    x2  4-  2a£  +  5  =  o,    x  =  —  «±  V«2  —  b. 
Pm  =  n  (n  —  1)  (n  —  2)  ....  (u  —  m  +  1). 


P„  =  \n . 

P, 

Cm  =  -J 


w  («  —  1)  (:<i  —  2)  ....(»  —  m  4-  1) 


-         I  m   "  |m 

^  («.r)  =  «ffo. 

r?  («#  —  bx  4-  c)  =  r?f/^  —  fofo  =  («  —  #)  die. 
rf  (.ryz)  =  a^flfe  4-  xzdy  +  f/zefo. 
,  /^\  _  y^  —  #^.y 

\yl  ~  y2 

d  (xn)  =  nxn~^  dx. 

d  (log,  x)  =  -~ 

tf  (Ioga  .t)  =  Jfa  —  • 

ge(i  +  x)  =  x 1 f-  etc. 

2        3        4 

/           X2        xz        xi  \ 

loga  (1  +  x)  =  Ma  [x  -  -  4-  : 4-  etc.). 

V  2  1  4  / 


l0ge  ( I    4-  2)  —log,  2 

=  2  (27TT  +  3  (a*  +  i)3  +  sW+^f  +  6tCT 

jf„  = i 

a—i—^(a—i)2+i(a-i)s-l(a—i)i  +  etc. 


logo  a 


_  x—  1—  $(.r—  i)2  4-|(,r— 1)3— ifo— 1)^4- etc. 
-  «—  1  — I  («  _  1  )2  + 1  (a  —  1 )»— f (a  I^+etc. 


304  F  O  li  M  U  L  A  S 

nt  r       nt  (nt  —  i)  r2 


/         nt  r       nt  (nt  —  i)  r2         ,    \ 

2i.     a=pii-\ + — i.-2  +  etc). 

1   \  in  i  •  2  ft2  / 

And  when  w  =  co  , 

22. 


/                  ^r2       ^3;*3  \ 

:.     a  =i? (i  +  tr  +  -, h  -r-  +  etc.) 

=  ppfr  =  jtf  (2.718281)^'. 

For  difference  series, 

.   7        (n  —  1)  (w  —  2)  7 

23.  an  =  «!  +  (w  —  1)  «i  + — di  +  etc. 

n  (n  —  \)  7       n(n  —  i)(n  —  2)  . 

24.  Sn  =  na,  -\ ^_    —Ldx  -\ * -^ ;  d,  +  etc. 

1-2  1-2-3 

For  eqnidifferent  series, 

25.  cvn  =  a,  +  (n  —  1)  d. 

a,  +  a, 

20.       0„  =   M. 

2 

7       a„  —  a, 

27.  a  = ■■ 

ra  —  1 

For  equimultiple  series, 

28.  a„  =  a1ww_1. 

a,  (mn  —  1) 

2Q.        On    = • 

m  —  1 
For  decreasing  equimultiple  series, 
30.     am  =  a,™00  =  o. 


31-     Sn  = 


m 

a 


in  (m  +  jj)  (m  +  2jj)  ....  (?>?,  +  77?)  — 

1  /  a a \ 

rp\m  (m+p)...  [m+(r—i)p]       (m+p)(m+2p) . . .  (in  +  rp)J 

33-    if   yz  +  py  +  q  =  ©, 

34.  -  =  00  ;     —  =  o ;     -  =  a,  or  indeterminate. 

O  00  O 

35.  -0   =    -2   =    -*   =    -6   =    _*»   =    +• 

,5_  1    —     3    _     _5    ___7—-     !«+!     __     < 


NOTES 


The  following  brief  sketches  of  eminent  mathematicians 
who  have  made  valuable  contributions  to  our  knowledge  of 
the  subjects  treated  in  this  volume  and  to  whom  reference 
has  been  made,  are  drawn  from  the  most  reliable  sources. 

Note    I.     (P.  104.) 

Neirton,  Sir  Isaac,  an  illustrious  English  philosopher 
and  mathematician,  born  at  Woolsthorpe,  in  Lincolnshire,  on 
the  25  th  of  December,  1642  (old  style).  He  entered  Trinity 
College,  Cambridge,  as  a  sub-sizar,  in  June,  166 1,  before  which 
date  it  does  not  appear  that  he  had  been  a  profound  student 
of  mathematics.  It  has  been  said  that  he  commenced  the 
study  of  Euclid's  Elements,  but  he  found  the  first  propositions 
so  self-evident  that  he  threw  the  book  aside  as  too  trifling. 
In  1664  he  discovered  the  Binomial  Theorem,  in  1665  took 
the  degree  of  B.  A.,  and  probably  in  the  same  year  discovered 
the  Differential  Calculus,  or  Method  of  Fluxions,  as  he  called  it. 

It  was  in  the  autumn  of  the  same  year  that  Newton  con- 
ceived the  idea  of  universal  gravitation,  the  suggestion  coming 
from  the  fall  of  an  apple.  It  would  exceed  the  limits  of  this 
notice  even  to  mention  his  many  remarkable  works  in 
Philosophy,  Astronomy  and  Mathematics. 

Near  the  end  of  his  life  he  said,  "  I  know  not  what  I  may 
appear  to  the  world,  but  to  myself  I  seem  to  have  been  only 
like  a  boy  playing  on  the  sea-shore  and  diverting  myself  in  now 
and  then  finding  a  smoother  pebble  or  a  prettier  shell  than 
ordinary,  whilst  the  great  ocean  of  truth  lay  all  undiscovered 
before  me."  He  died  at  Kensington  on  the  20th  of  March, 
1727,  and  was  buried  in  Westminster  Abbey. 


306  NOTES. 

Note    II.     (P.  186.) 

M'Lanrin ,  Colin,  a  Scottish  mathematician,  was 
born  in  Kilmodan,  Argyllshire,  in  Feb.  1698,  and  died  in 
Edinburgh,  June  14th,  1746.  He  was  a  graduate  of  the 
University  of  Glasgow,  and  in  17 17  was  appointed  Professor 
of  Mathematics  in  Marischal  College,  Aberdeen,  which  posi- 
tion he  held  till  1725,  when  at  the  recommendation  of  Sir 
Isaac  Newton  he  was  called  to  the  mathematical  chair  of 
Edinburgh.     He  held  this  professorship  for  over  twenty  years. 

His  principal  works  are,  "  Geometrica  Organica"  "A 
Treatise  on  the  Percussion  of  Bodies,"  "  On  Fluxions,"  said 
to  be  the  most  complete  treatise  on  the  subject  and  the 
author's  most  profound  work  ;  "  A  Treatise  on  Algebra,"  and 
"An  Account  of  Sir  Isaac  Newton's  Philosophical  Discoveries." 

Note    III.     (P.  191.) 

Napier,  John,  Baron  of  Merchiston,  was  bom  at 
Merchiston  Castle,  near  Edinburgh,  Scotland,  in  1550.  He 
was  educated  at  the  University  of  St.  Andrew's,  and  is 
celebrated  as  the  inventor  of  Logarithms.  His  loga- 
rithmic tables  were  first  published  in  1614  under  the  title 
"  Mirifici  Logarithmorum  Canonis  Description  Napier  also 
enriched  the  science  of  Trigonometry  by  the  general  theorem 
for  the  resolution  of  all  the  cases  of  right-angled  spherical 
triangles.     He  died  in  16 17. 

Note    IV.    (P.  191.) 

Brir/f/s,  Henry,  an  eminent  English  mathematician, 
born  at  Warleywood,  near  Halifax,  about  1556.  He  was 
educated  at  St.  John's  College,  Cambridge.  In  1596  he  was 
chosen  Professor  of  Geometry  in  Gresham  College,  London. 
He  became  first  Savilian  Professor  of  Geometry  at  Oxford 
in  16 1 9. 

He  is  chiefly  distinguished  for  the  improvement  and  con- 
struction of  logarith  ms.  No  sooner  was  Napier's  system  of  Loga- 
rithms published,  than  Prof.  Briggs  began  the  application  of  the 
rules  in  his  " Imitatio  Napierea."     He  greatly  improved  upon 


NOTES.  307 

Napier's  plan,  by  adopting  10  as  the  base  of  his  system,  and 
he  has  the  honor  of  being  the  author  of  the  system  now 
in  general  use.  He  published  in  1624  a  work  entitled 
" Logarithmica  Arithmetica"  containing  the  logarithms  of 
all  integral  numbers  to  20000,  and  also  from  90000  to  100000, 
calculated  to  fourteen  places.     He  died  in  1630. 

Note    V.    (P.  263.) 

Sturm,  Jacques  Charles  Francois,  an  eminent 
Swiss  mathematician,  was  born  at  Geneva,  in  September,  1803. 
He  was  tutor  to  the  son  of  Madame  de  Stael,  with  whom  he 
visited  Paris  in  1823.  In  1827  Sturm  and  his  friend  Colladon 
obtained  the  grand  prize  of  Mathematics,  proposed  by  the 
Academy  of  Sciences  in  Paris,  for  the  best  memoir  on  the 
compression  of  liquids.  He  discovered  in  1829  the  celebrated 
theorem  which  bears  his  name.  He  became  Professor  of 
Mathematics  at  the  College  Eollin  in  1830,  a  member  of  the 
Institute  in  1836,  and  Professor  of  Analysis  at  the  Polytechnic 
School  in  1840.     He  died  in  1855. 

Note    VI.     (P.  267.) 

Homer f  TV.  6?.,  was  an  eminent  English  mathemati- 
cian, born  near  the  close  of  the  last  century.  He  was  a 
teacher  of  mathematics  in  Bath,  and  died  in  1837. 

About  fifty  years  ago  he  discovered  the  Method  of 
Synthetic  Division,  otherwise  known  as  the  "Method  of 
Dividing  by  Detached  Coefficients."  In  1819  he  com- 
municated to  the  Royal  Society  his  method  of  solving 
algebraic  equations  of  all  degrees,  entitled,  "  A  New 
Method  of  solving  Numerical  Equations  of  all  orders, 
by  continuous  Approximations."  Previous  to  this  time 
there  was  no  direct  and  reliable  method  of  finding  the 
roots  of  equations  beyond  the  fourth  degree.  By  his 
method  the  process  is  comparatively  brief  and  simple. 

This  method  is  regarded  as  among  the  most  valuable 
contributions  to  the  science  of  Mathematics  in  modern 
times.  The  first  elementary  writer  that  saw  the  value 
of    it,    says    De    Morgan,    was    Prof.    J.    R.    Young,    who 


308  NOTES. 

introduced  it  into  his  Treatise  on  Algebra,  published 
in  1826.  Prof.  Young  says  it  is  the  shortest  method 
of  extracting  roots  of  higher  equations  that  he  has  seen. 

Note    VII.     (P.  282.) 

Cavdan,  Jerome,  an  Italian  physician  and  mathema- 
tician, born  at  Pavia  in  150 1.  He  graduated  as  doctor  of 
medicine  at  Padna  in  1525,  and  was  successively  professor  of 
mathematics  and  medicine  at  Milan  and  Bologna.  He  dealt 
much  in  Astrology  and  was  a  professed  adept  in  magical  arts. 
Among  his  numerous  writings  are,  "Ars  Magna,"  "De 
Rem  in  SuMilitate,"  "  De  Rerum  Varietate,"  "  De  Vita 
Propria?  and  several  medical  works.  In  1545  he  published 
in  his  "Ars  Magna"  a  method  of  solving  cubic  equations, 
now  known  as  "  Cardan's  Formula."  He  was  the  first  that 
noticed  negative  roots.     He  died  at  Eome  in  1576. 

Note    VIII.    (P.  286.) 

Desert rtes9  Rene,  (Lat.  Renatus  Cartesius)  an  illus- 
trious French  philosopher  and  mathematician,  born  at  La 
Haye,  in  Touraine,  March  31,  1596.  He  was  educated  at  the 
College  of  La  Fleche.  On  leaving  college,  at  the  age  of 
nineteen,  he  resolved  to  reject  all  scholastic  dogmas  and  to 
free  himself  of  prejudices,  and  then  to  receive  nothing  that 
was  not  supported  by  reason  and  experiment.  To  perfect  his 
education  he  determined  to  travel,  and  to  this  end  entered 
the  Dutch  army  in  1616  and  came  into  the  service  of  the 
Duke  of  Bavaria  in  1619.  I'1  1620  he  was  in  the  battle  of 
Prague,  but  soon  renounced  the  military  profession  and  gave 
himself  to  more  congenial  pursuits.  In  1637  he  produced 
his  celebrated  "Discourse  on  the  Method  of  Reasoning  Well 
and  of  investigating  Scientific  Truth?  in  which  were  included 
treatises  on  Metaphysics,  Dioptrics  and  Geometry.  This  last 
treatise  included  the  method  now  known  as  the  Cartesian 
Geometry.  The  formula  given  in  our  Appendix  for  the 
reduction  of  biquadratic  equations  is  due  to  him.  He  died 
at  Stockholm  in  February,  1650. 


ANSWERS 


Page  11,  Art.  16. 

i .  The  square  of  the  side. 

2.  Twice  the  radius. 

3.  The  circumference  is  3.1 416  times  the  diameter. 
The  area  is  .7854  times  the  square  of  its  diameter. 

4.  Interest   is   the   product  of  principal,   rate   and   time. 
Amount  is  the  product  of  principal,  1  +  rate  and  time. 

5.  The  product. 

6.  The  product. 

Page  26,  Art.  92. 


I.    — . 

7- 

+  • 

r  1 .   Has  no  sign. 

2.     — . 

8. 

+  . 

12.    4-. 

3.  a  is  —  and  b  is  +. 

9- 

— . 

13-    +• 

4-6.  Given. 

10. 

+  • 

14.  Has  no  sign. 

or 


Page  27,  Art.  96. 

2.  (a  +  by  =  a2  +  2ab  +  b2. 

3.  (a  —  by  =  a2  —  2ab  -f-  b2. 

4.  (a  +  If  +  («  -  by  = 

2  {n2  +  b2). 

5.  a2  —  b2  ~-  a  —  b  =  a  +  b, 


a2  —  b2 

=  a  +  b. 

a  —  b 

x$  —  y*  1         1 

6.     — -^   =  0*  _  y\. 

X*   +  2/T 

7-   (*  +  */)2  -  (x  -  yf  = 


4xy. 


5.  10. 

6.  4. 


Prtr/e  28,  Art.  98. 

7.  ±  2. 

8.  +  2. 


9.   144. 
10.   16. 


310 


ANSWERS, 


i.  Given. 

2.  i  ia  +  Sx  —  2a2  +  ax. 

3.  a--  +  44*  -\-2c1b-  —  ab. 

4.  6a2b  +  ioai2  —  Sax2  —  a2x. 

5.  «Z<2  —  zulb  +  a3  —  6ac 

—  4ttC2  —  c3. 

6.  2>xy%  +  9r2i/  +  7x2y2- 

7.  #z  —  2to  4-  ex. 


Page  31,  Art.  100. 

8.  3«*#2  —  «5s  —  asb. 

9.  aa^y  —  2ax\f. 

10.  Vei  +  V«c. 

1 1.  3a2Z>  —  aJ  +  2b, 
or  J  (3ft2  —  a  +  2). 

12.  r/2(3.r  +  2«  —  35). 

13.  n  (m  —  2d  4-  b). 


14.  Given. 

15.  (a  +  £)*  —  (a  — J)* 


P«</e  32,  Art.  110. 

16.   (tf  —  b  —  c)  Vx  —  1. 


1 .  4ax. 

2.  2x%  —  2>x  +  7« 

3.  5«J. 

4.  2«2  +  ax. 


17.   Va  —  #• 

Page  33,  Art.  114. 

5.  {a  —  1)  (3  +  y)      9.  o. 

-  2%- 

6.  —  2  Vz2  +  a2. 

7.  2#  —  2C. 

8.  6z8  +  to2. 
Page  35,  Art.  110. 


10.  o. 

1 1 .  40b. 

12.  2  (b  —  c 

13.  —  62. 


m). 


1.  2rt —  2J — 2c  —  42+21/ —  4. 

2.  3fl  _  3&  _  3c  +  3to  —  9. 

3.  2a  —   2C. 

4.  2£  (a  +  J  —  c). 

i7 1  1         11,11 

5.  aatoca  _|_  2%?o*c«, 

or  (2*  +  i)a^jM, 

6.  2a2  (c  —  5). 


7.  ZCESb*  —  Xlf. 

„  1,1  S ,  J  , 

8.  2a-0-   —   2rt-"'//S  —  rt0. 

9.  —   2C  V#  +  #• 

10.  4a- ^. 

11.  2«  +  //  +  &  =  2rt, 
or  2ff  +  2b. 

12.  4to?  —  2am. 


Page  38,  Art.  130. 


1,  2.  Given. 

3.  \a3x2yH. 

4.  ^tocz3. 

5.  2ia*bx*y8. 

6.  5r<w+7'+1to+*+1a;z. 


+  wi  +  2  — +ra+S 

7.  DO"  0m 

8.  3a2to. 

9.  4. 

10.  a^J2*. 

11.  a^b2™. 


ANSWERS. 


311 


12.  aimb2m. 

13.  zMtf1. 

14.  x2>^y. 


'2n-—Sn  ,i2?n-—m 


15.  x*"—*'ly 

16.  6aibx'-. 

17.  =F«4Z/4.r>. 


18.  —   2alb2X2. 

,      J      10 

19.  oa3x'»'. 

is.    18 

20.  a? a; * . 


J'rtf/e  39,  Art.  131. 


21.  Given. 

22.  «4  —  2a2£3  +  z4. 

23.  2azb  —  2a2bx2  —  20b3. 

24.  a5  +  ft4.?  —  ax*  —  x5. 


25.   2r<2a;  +  2abx2  +  2«ca;3 


—  axA 


bx3 


ex*. 


26.  4«2  —  b2x2  +  2&ra3  —  c'x*. 
2  7 .   «3ot  _  amffln  _(_  fl2»i£n  _  £3ra 

28.     I   —  2A'2  4-   2X3  —  2a;5  +  X6. 


Page  40,  Art.  133. 

30.  3a7J  —  $a%b2  +  2a5bs  +  7 r^4/>4  —  ja3b5  —  5«2J6  +  5^ 

31.  a5  —  7,a3x2  4-  a2x3  +  2f/.r4  —  a;5. 

32.  .r6  4-  x*y2  —  x2xf 


//" 


33-  (ts  —  atx  —  5«4£4  +  8a3x5  —  4«2£6  +  ax7. 


Page  44,  Art.  144. 


5.  ^ab  3c  hi2,  or 


4ad2 

W 


6.  \a  ^b~hd~2,  or 


ytbd2 


7- 


8. 


$a2c5d2 
~~b3 


9.  Given. 

10.  —  4f<w-1Z<"-1. 

11.  —    2*74Z>2. 

SaP 
~bn 


b2z« 

Page  45,  Art.  145. 

4-z.i 

15.  —  yt*o*. 

16.  ab?. 
a 


12.  57"'//-",  or 

13.  —  4tfZ<2. 


14.   —  \a~xb~2,  or  — 


4ab2 


17.  —  j  =  —  ab~\ 

18.  —  am"-bnl 

19.  —  am;i-mbn'1-n. 

20.  tfi+lymr-\ 

21.  an+2bn  +  ixn. 


Page  4(>,  Art.  140. 


1.  J.r  —  2f?.r2  4-  ytV/x3. 

2.  2   4-  rt"Z/2m  _  3ft3»J2»n# 


3.  -2f/w— " 24-  Aa2m~ nb2n~ m. 

4.  2ambs~2n  —  3<7w  +  2Mfr!. 


312 


A  NSWERS, 


5.  2;tm-2nbs  —  3ambs+2a. 

6.  2a~m-»]u* '-»)  —  3<i2mb~n,  and  2a~nb-n  —  ytubn. 


7- 


j"-1      y*^-  _         j     j  _ 


# 


=  a*-^ 


1  .--1 


IT  +  _ 
0        a 


+ 


1,  2.  Given. 

3.  a4  —  8./;2  -J-  42  —  1. 

4.  a;6  +  xHf  +  jbV  +  >/. 

5.  a4  —  e&c2  -j-  "4- 

,3         11         11         3 

6.  a*  +  ft-a'  +  (fix*  +  a* 

7.  ft  4-  ft-a;2  —  x. 

8.  aw  +  a11. 

9.  ft"  6"  —  an-W\ 

_a  -I7-1  7—2 

10.  ft    a  —  ft    36    3   +   0    * 

11.  ft3  —  5«2a;  —  «£. 


I'ftf/e  48,  Art.  148. 
12.  a*. 


13.  I    —   2£  +  X2. 

14.  &2  +   26a  4-  x*. 

15.  a3  4-  30%  +  3-^2  +  9/ 
32i/4 


+ 


W 


2ft— 13 


16.  2ft2 — 2ft — 2  4- 

$<t2— 2ft  4- 1 

17.  2ft2  4-  $az  4-  a-2. 

18.  Given. 


1.  Given. 

2.  ft2  —  2ax  4-  a;2. 

3.  ft  —  J. 

4.  ft2  —  ax  +  a2. 


P«£/e  50,  .4r£.  iJO. 

5.  a:3  —  3a-2  +  3./:  —  I. 

6.  a,-3  —  2.r2_y  4-  2yz. 

7.  1  —  2b  4-  3Z/2  —  4J3. 

8.  2  ft2  4-  5ft.r  4-  a2. 

Page  51,  Art.  151. 


5.  Given. 

6.  ft4  4-  (fib  4-  ft63  4-  bi 

7.  x2  4-  2.r,y  4-  if. 

8.  ft3  4-  3ft2a;  4-  ytx2  -\-  xs. 


Page  52,  Art.  151. 

12.  a;5  4-  a4  —  2a3  —  2a2. 

13.  xA  4-  10a3  +  30a2  +  87a  4-  268  4-  -7-9-^-. 

*  —  3 


14.  a8  —  a,a  —  2a;  4-  4 Also  a3  4-  8. 

x  —  2 


ANSWERS. 


313 


15,  a4  —  x3  —  4X2  +  Gx  —  6 


x  4-  1 


16.  x5  —  3-r4  -f  4a"  —  12a;2  +  36.C  —  106  +  -$1'—. 

x  +  3 

17.  a:5  —  3a4  +  3:t3  +  3^  —  3X  +  1  ', 

Also  a,-5  —  5a4  4-  u^3  —  na;-  4-  52;  —  1. 

18.  iC4  —  2a;3  —  -ix2  +  6x  —  6  H —  ; 

d  z  4-  2' 


Also  a"4  4-  3a;3  4-  22'2  4-  6a;  4-  24  4- 

19.  a;3  4-  x  4-  1  ;  Also  ,t2  —  2./:  4-4  — 

20.  ./'6  4-  a-5  4-  xi  4-  .r3  4-  x2  4-  .r  4-  1  ; 


*  —  3 

a;  4-  2 


Also  x6  —  x5  4-  a4  —  a;3  4-  a;2  —  a;  4-  1  — 


x  4-  1 


21.  ft8  —  «4a^  4-  xs ;  Also  «9  —  cfix3  4-  a3a;6  —  a;9  4- 


2-r1 


a3  4-  a;3 

.10 


2?/1 


22.  xs  —  :r67/2  4-  a*y  —  a%6  4-  ?/8 ;  Also  x5  —  y5  + 


23.   'l*  —  a40  4-  xs  —  x6  4-  a;4  —  x2  4-  1  ; 
2 


,•12 

And  a;7  —  1  - 


24.  a;i3  _  rt%6  4-  a24 ;  And 

,i6_f/4ii:i4  +  ft8i?.i3_ rti2A.io  4.  a163?—a?°zfi 4- a24.r4— rt28a-2 4- a32. 

25-  •  6  —  aV  +  2/8  5  And  ((i  — a^  +  a%l)i  — a^  +  ^8- 


26.  a'4  —  33^  —  4a;2  4-  19a;  —  61 


174 

x  —  3 


And  a4  —  93^  4-  32a;2  —  103a;  4-  305  4- 

27.  x6  —  x5  4-  4a4  —  6x3  4-  a*2  —  x  4-  4  — 
And  a^  4-  a;5  4-  4a4  4-  2a:3  —  ^x2  —  3a:  4- 

28.  x5  4-  3a:4  —  io.r3  4-  12a;2  —  19a;  4-  23  — 


924 

s  4-  3 

3      . 

*  4-  1  ' 

1 

—  1 


a: 


7 


a-  4-  1 


And  x5  4-  5a4  —  2 x3  —  jx  —  3  4- 


13 
x  —  1 


[14 


ANSWERS. 


Page  ~>.~>,  Arts.  1Z5-15U. 

To  give  the  answers  to  problems  in  Factoring  Avould 
destroy  their  value  to  the  student.  They  are  therefore  omit- 
ted. The  same  reason  may  be  inferred  when  other  answers 
are  omitted. 

1'aae  fil,  Art.  ItiS. 


i,  2.  Given. 

3.  a  (x  +  a) 

4.  x  +  5. 

5.  b  (a  +  b) 

6.  x  {a  4-  b) 


7.  a  —  b. 

8.  x  —  1. 

9.  a  —  x. 

10.  (x —  i)2  (x  —  2). 

11.  X   +    2. 


12.  3 

13-  3  +   3- 

14.  X  +   2. 

15.  £2  —  4. 


IVff/e  62,  Art.  173. 


1.  a4  +  «.t3  —  ff3.r  —  <v4. 

2.  Gahtfy5. 

3.  a3  4-  <y,V  —  a#2  —  a;3. 

4.  at3  +  a2  —  x  —  1. 

5 .  a4  —  x%  —  x  4-  1 . 

6.  xm+nym^n. 

7.  a4  —  2a2  4-  1. 


8.  «4 
9 


«3a  +  a 


10.  <74  —  2ft2a2  4-  x*. 

11.  a4  —  1 . 

12.  a10  —  a9?/  4-  2:i8y2  —  x"if 

+  X*y*—xhf-\-xhj 

13.  as  —  ahfi  4-  <76a2  - 


2.<2ys 


14.  a,4  4-  2.r; 


9x> 


2X  + 


15.  a4  4-  5^'3  +  5X~  —  5^  —  6- 


a4  4-  flfte  —  ff-c3  —  a4. 

JV/f/e  6*6,  .4r/.  193 

a2  —  xy  4-  y~ 


Given. 

6a*d?e[ 

2 
yibc 


2yz. 


x  —  y 

ri  _  x2  +  1 


A'2  —    I 
I 


X*  _  ^2   4.   yi 


«  +  3 

2 
3"(^+7)' 
a  —  a 
x  4-  « 


1  — a 

3.  r'2  4-  CO  +  &*• 

4.  „8 — a6&2  4-  a4*4 — fc8&6  +  W 


5.   r?12  —  rt6£6  +  J12. 

rr>2  -f-  a. 

x  4-  1 


6. 

7.  x  +  : 

8.  £2  + 


x3  +  a-  —  1 2 
J  -  1 

a  +  T 


ANSWERS. 


315 


a 


x 
ab 


a4  +  a2  —  2 


Page  67,  Art.  195. 

2.V2    -)-    2 


.r  +  i 
«  —  a; 


6. 


2rt5 


7.  o. 


a2c 

— 

c 

a6 

— 

2fl4 

+ 

2  a2 

— 

I 

a6 

+ 

1 

a4 

+ 

2  a3 

+ 

2  ft2 

+ 

2  a 

+ 

a*  —  1 


Page  68,  Art.  196. 

s6  +  5^5  +  5y  +  25 

a;2  —  2.T  —  35 
ff3  +  a2  +  « 
«  +  1 


6.     „- 

xi  —  1 


Page  69,  Art.  19 S. 

x*  —  xsy  +  x2y2  —  xys  ^    x4  +  x2y2  _ 


X*  —  y4 


'      re4  —  y4  '    a4  —  y4 


a4  +  1  ^    2a4  —  2  _    3a6  —  3a4  +  3. 


,.2  _ 


2a  —  1 


4  (a7  +  a6  —  a  —  1 ) 


8  («8  +  a6 


1)' 


2  (a7  —  a6  4-  2a5  —  2a4  4-  2«3  —  2«2  4-  a  —  1)  m 
8  (a8  +  a«  _  a2  _  !)  _5 


8  (a8  4-  a6  — a2—  1) 
4  (x2  —  4)        x3  —  3a2  +  4  #    x5  4-  3a2  —  4 
4-  34  _  5X2  +  4  5   34  _  5^qT"4 '   ^  —  5^4-4" 

5.   C.  D.  is  (x4  -  1)  (x3  4-  3^  _  3x  4-  3)  (^2  +  3). 
Numerators  are 
a  (a3  +  t,x2  —  33  4-  3)  (z2  +  3) ; 

(*2  +  i)2(^-i)(^2  +  3); 

And  (a4  —  1)  (a;3  4-  3a2  —  3a  +-3). 
s  +  y  .    (x  —  y)  O2  —  ?/3)  _    (.r2  +  ?/2)  (a2  +  y3) 


a4-?/ 


t6' 


f 


//< 


316 


I, 

2.  Given. 

3«  («  +  x) 

3- 

¥ 

4- 

2  (a  -\-  b) 
a  —  b 

5- 

o. 

6. 

—  i. 

ANSWERS. 
Page  70,  Art.  109. 

x  +  4 
a;  —  4 

8    -*_ 

'  a;  +  rt2 


9-  " 


—  3 


xd  +  2x2  —  9.r  —  il 
-a-2 

a;4  -f  x3  —  x  —  i 


x 


a  +  x 
i 

x  +  r 

i 


a; 


a 

X 

X 

a 

+ 
3 

W- 

X3 

i 

i 

>'    (a8  _  £3)2 


3^ 

2XI/Z2 
9«<  2d 

a:2—  3a;  +  9 
a;2  —  4 


Page  71, 

4-  5- 

^«. 

20i 

1 

a;  +  4 
*  — 3* 

6.  -5-. 

.2  +  4 

'"  a;2  — rt5 

s           IO 

x%  —  ax4  +  d' 

Page  72,  Art.  202. 

5- 
6. 


a;6 

— 

cfi.r3 

+ 

a4 

a^ 

— 

a3x2 

+ 

ac 

z5 

+ 

a3 

x3  +  a5 
aa:  +  7a;  +  4a  +  28 
«.r  +  5a;  +  6a  +  30 
a  —  1 


(x*  —  x  +  1)  (a2  —  a  +  1) 

9.   —  (a;5  +  x*  +  a;3  +  a;2  +  a;  +  1 ). 


IVj/e  72,  Art.  203. 


x 
4-c' 


4  +  a 


J    (1  -  O  (3  -  «2) 

a2  +  ay  +  if 

x2  —  f 


ANSWERS, 


317 


Page  73,  Arts.  205,  206. 


25a2  —  $ab  _    25a2 
acex 

c10  +   C5X6  _|_  xi% 

a1  —l$x%  4-  x*  ' 


5a.      2.  3^  (1  —  x). 

b*  (c3  +  d3) 
5'  a  («62  +  c2«") ' 

,    Q/8-y4+  i)(a+  1) 
h  a*  +  ^  +  1 


#4  (a;12?/12  —  z10?/10  +  z?ys  —  a-6?/8  +  a-4?/4  —  xhp  +  1 ) 
y6  (//6  —  rt#5  +  a^yi  —  a3?f  +  fl4#2  —  a5  +  «6 


c   —  .Sc 


14 


c2  —  169 


rr  —  4 


8a  4-  16 


a4  —  y5 


Page  75,  Art.  210. 


a3  -  a*~b  +  sab2  4-  53 
a  (aaTTjs) 

2a2  —  2a  -f-  1 
a  (a  —  1) 

az 
a  4-  # 

a4  4-  6a2  +  1 


a  —  1 

x1  +  ax  4-  a2 
2;  —  a 
2a# 


a  +  ic 

a  —  x 

ab 


y- 

x  —  1 

10. 

a2  —  x\ 

11. 

a  +  x. 

12. 

aW  —  a2  - 

-i84-  1 

«2- 

&2 

13- 

1. 

14. 

1 

x 

X 

J5- 

a. 

16. 

I+^2' 

17- 

a  —  1. 

18. 

1 

-5.  Given. 
x  =  12. 
*  =  7. 


Pagre  82,  Art.  233. 


abc 


9.  x 


ab  +  ac 
J  -c 


be 


318 


ANSWERS. 


mn  —  ab 

io.  x  =  — ■ — i 

a  +  o  —  m  —  n 

_  tfc  +  al?  +  be*  —  (a  +  b  +  c) 
ab  +  ac  4-  be  —  i 

1 6.  x  =  4. 

17.  #  =  6. 

18.  x  =  4. 

19.  a;  =  7b. 

20.  a;  =  ^6  (1 — a2). 


12.  2  =  5. 

13.  x  = 


m  —  n 


14.  x  =  5. 

15.  2;  =  7. 


C  +  2(tf  —  J) 

21.  K= S -. 

zb 

22.  2;   =    2. 


PROBLEMS. 

1.  2  dols.,  6  halves,  30  qrs.,  90  dimes,  and  450  half  dimes. 

2.  36  years. 

3.  210  acres. 

4.  -       -  =  the  greater,  and =  the  less. 

2  2 

c.  — =  one  part,  and  — =  the  other  nart. 

0  2  x  2 

aw  —  c»  +  J       ,  era  +  «  —  J 

6.  The  parts  are and 

11  +  1  «  4-  1 

7.  $80000. 

8.  —  $600. 

9.  1st,  6  miles ;  2d,  15  miles  ;  3d,  00  miles. 

Page  89. 

i7- 


10.  a  years 

11.  $93. 
52  8o??^c 


SZOU7W  ,       .  „ 

12.  — ^ ;  c  being  feet. 


13 
14 


5280m  —  nc 
s 


1  +  a 

abc  , 

— =-  days. 

r/Z>  +  ac  +  fo      J 

15.  40  gallons. 


16. 


c-b 


a 


a 

n  —  m 

a 


m  4-  11 

19.  Brandy  30.  wine  40,  and 

water  70  gallons. 

20.  A's,  ^8si;  B's,  Igif ; 

C's,  $83^-. 

21.  30  and  7.     Prob.  4  gives 

the  formula. 


ANSWERS.  319 

Page  'JO. 

22.  60  apples  and  20  oranges. 

23.  x  =  —     —  ;  in  which  x  =  original  number  of  oranges, 

J  m  —  n  °  ° 

mx  =  the  number  of  apples,  a  =  number  of  apples,  and  b  = 
number  of  oranges  sold,  and  n  =  ratio  of  oranges  to  apples 
after  the  sale. 

2a  (1  —  mn)        „  .      ,, 

24.  — =  No.  sold  m  all. 

mn  +  m  —  2 

—  — -   =  No.  of  apples  at  first. 

mn  +  7U  —  2 

— —  =  No.  of  oranges  at  first. 

mn  +  m  —  2 

25.  1st,  140  qts.  ;  2d,  60  qts. ;  3d,  45  qts. ;  4th,  80  qts. 

26.  70;  25  ;  36  ;   15  ;  20. 

27.  1^  of  a  mile  per  hour. 

28.  1  mile  per  hour. 

29.  1000. 

Page  95,  Art.  251. 

1.  Given. 

2.  x  —  2,    y  =  3. 

3.  x  =  12,     ?/  =  18. 

5.  a  =  11,     y  =  9. 

//  —  bd  ad  —  h 

6.  x  =  r,     y  = r- 

a  —  b       J         a  —  b 

7.  x  =  7i?  (a  +  b),     y  =  h  (a  +  V)- 

8.  x  —  7L     y  =  5. 

9.  a:  =  244,     y  =  —  172. 

10.  2;  =  8,     y  —  12. 

4a  +  b  b  —  2Ci 

11.  x  =  ^-z — ,     y  = 

6  J  12 

b'r  —  be  _  «'c  —  ac' 

l2-  x  =  ah>  _  a>b>     V  —  a'b^-~ab'' 

13.  The  equations  are  not  independent  and  represent  but 
one  condition;  viz.,  that   y  =  \y. 


320  ANSWERS. 

b'  —  b  ab'  —  n'b 

14.  x  —  ;,     V  —  —j— 

a  —  a'      J  a  —  a 

15.  x  =  oq,  y  ==  00.  The  two  conditions  are  incompatible 
except  for  infinite  quantities. 

16.  x  =  3,     #  =  4. 

17.,  =  —-,    y  =  -j- 

18.  x  =  6,     y  =  4. 

19.  x  —  4,    y  =  15. 

Page  96. 

PROBLEMS. 
i.  The  number  is  462. 

2.  6  oranges  and  10  apples. 

3.  The  fraction  is  -ff. 

4.  The  1st  in  12,  the  2d  in  20,  and  the  3d  in  30  hours. 

5.  The  problem  is  indeterminate,  only  one  condition  being 
given  from  which  to  determine  two  unknown  quantities,  viz., 
that  the  interest  on  the  sum  for  2  months  shall  be  §10. 

6.  A  in  25  days  and  B  in  i6|  days. 

7.  A,  I ;  B,  -j%  ;  and  C,  ^.     C  worked  6  days. 

_    ac  —  ab    ,  ,,         ,        .  ab  —  be 

8.  —r  01  the  1st,  and j-  of  the  2d. 

a  —  0  a  —  b 

Page  07. 

9.  Loaves  6  cts.  each  ;  Fruit  20  cts. 
10.  The  fraction  is  f. 

(;»2  +  mn)  c  —  («2  +  ab)  p 


11.  1st 

2d, 


(a  -f  £)  (w  +  n)  (nib  —  an)  ' 

(M8  +  WMl)  C  —  (62  4-  tfJ)p 


(a  +  b)  (m  4-  »)  (««  —  mb) 

12.  The  problem  is  indeterminate.         / 

13.  $400  and  $100,  at  2%  and  4%. 

14.  The  fraction  is  ^. 

15.  A's,  500  dollars  ;  B's,  —  500  dollars. 

16.  A,  $900  ;  B,  $600.     The  mortgage  $500. 


ANSWERS.  321 

Note. — The  student  will  observe  that  the  preceding  problem  contains 
really  two  independent  problems. 

Page  09,  Art.  252. 

i.  Given. 

2.  x  =  3,     y  =  2,  and  z  =  i. 

(a2  —  #c)  w  +  (/>3  —  rtc)  »  +  (c2  —  a b)  r 


3.  x  — 

y  = 


a3  +  b3  4-  c3  —  3«#c 
(£2  —  «^)  m  4-  (c3  —  <?#)  w  4-  («2  —  #c)  r 
a3  4-  63  4-  c3  —  ytbc 
_  (c2  —  a b)  m  4-  (a2  —  be)  n  4-  (&2  —  ac)  r 
a3  4-  b3  4-  c3  —  3«^c 
4-  a  =  .5,     2/  =  7,     *  =  4,     u  =  3. 

5.  a;  =  2,    y  =  3,     2  =  4. 

6.  a:  =  1,     y  ■=.  1,     z  =  2,     it  =  2. 

2abc  2abc 

ab  -{-be  —  ac'     "*  ~  ac  4-  be  —  aV 

zabc 

ab  +  ac  —  be 

8.  x  =  o,     y  =  o,     z  =  o. 

_  (a  +  b)  m  4-  cw  _  (a  +  b)  n  4-  cm 

~(«  +  bf  —  (*'     y  '      (a  +  b)*  —  (F' 

2ac  2bc  20b 

10.  x  = ,     y  = ,      z  =  -    — t- 

a  4-  c      J       b  +  c*  a  +  b 

PROBLEMS. 

1.  1257. 

2.  40,  6o,   70. 

3.  A's  age  =  3«  —  2c,     B's  =  2&  4-  2c  —  3a, 
C's  =  3«  —  2  J. 

Pajye  J  00. 

4.  40,   60,   50. 

5.  8122J} §  and  $97f|,  value  of  Horses  ; 
$32iL  and  $12^,  value  of  Saddles. 

6.  A,  §800 ;  B,  $900 ;  C,  $600. 


322  ANSWERS. 

. ,     _     m'm'W(n'— 1)(«— mn')—  mm'm"(n— n'){n'n".~ i) 
n"(m"n'— m')(n— mn')— mn"(m'n— n')(n'n"—  i)  ' 

-n,    _      nui'\ii—n)  (m"n' —m')—mni  n'\n—i)(m' n  —  n') 
u'\ii  —  uiu')(iri'>i—m')  —  iniH'\Hii,-^{jn^i^n')' 

p,    _     mn'ri  m'n"-  m"  [i  —  m')—n>i')i"(m'  —  m")(i—mm') 
m(m'n—m"J{ n'  —  m'n)— nn"(m'— m"nT)(i—  mm' )  ' 

8-  A>  --^-r-L7, 17/  5     B> 


ub  +  &rf  —  ad '       '  ad  -+■  bd  —  ab' 

p  2«fo/  ~  2  abed 

^>  7J7J   I    77       77J? '     ^' 


ad  -{-  ab  —  bd '        '  abd  —  acd  +  bed 
9.  A's,  $1000  ;  B's,  —  $500  ;  C's,  $0. 
IO-  3  ?  5,  and  6  miles. 

JV/f/r  106,  Art.  274. 

5.  ft2  +  2ax  4-  a:2  and  «2  —  202;  +  a;2. 

6.  4a2  4-  4«&  +  Z/2  and  «2  —  4^  4-  4b2. 

I         b  b*  .     ±        b  & 

7.  a~  -\ j -  +  etc.  and  a~ r s  —  etc. 

2tta         8«'3  20s         8«? 

8.  .r6  +  6.r5y  4-  15./'4//2  +  2o.r3//3  +  15/*!/  +  6xif  +  if; 
Xs—  7xey  +  2ix5f  —  35^y  +  35-''3i/4—  2ix2y5+jxy6— //<. 

9.  32a5  -|-  i6oaib  +  i,20(fib~  4-  320«2Z/3  4-  i6o«£4  4-  32^; 
27a3  —  8ia26  4-  8i^2  —  27J3. 


I  X  X 


•2 


io1. 4- .4-  etc. 

a       (fi       a3       ai 

I  2X         -zx*        AXS 

a3       a3       a*         «5 

I  2.T  X%  4.T3 

11.  a3  4 l -4  4-  — ^  —  etc. ; 

3a3       9«:!       8i«» 

I        22;  x2  AX3 

a* - -. —  etc. 

3«3       9a5       8if/J 

s  s  1  erf*  erf 

12.  a?  +  %a%  +  ^¥  4-^-^4-  etc. ; 

i6d*      64a2 

B  7     T  9  .111       CLU* 

a-       2a-      8«?        i6«*~ 


ANSWERS. 

10 5'" 


323 


1  s^        W2       1015c8 

i3-  T  +  ^r  +  fr-  +  — tt  +  etc. ; 
a*       2as      8a-        ioa~s~ 


0s  —  fare  +  J£-a2t2  — 


5^ 
i6a* 


5c4 


—  etc. 


14.  aV  4- 


+  etc.  ; 


by 


—  +  -f-  +  Hh  +  etc. 
«2^       2  a- £3       Srr-u-- 

15.  r/3  +  4^5  —  Sac  +  4^  —  \6bc  +  16c2. 

16.  4a2  —  \2ffix  +  4rt&2  —  $axy  4-  4«2  4-  9«2.e2  —  6ab2x 
-f-  6ea-3?/  —  6«.rz  +  Z/4  —  2#\n/  +  2#2z  4-  .r2*/2  —  2.^2  +  z2. 

17.  a2aj2  4-  2abxy  —  6axz  4-  ioax  4-  b  y2  —  6byz  4-  10 by 
4-  922  —  302  +  25. 

18.  a3  4-  3«26  —  3«2c  +  yf~d  —  6bcd  4-  b3  +  3«i2  —  3//^ 
4-  $b2d  —  6acd  —  c3  4-  yic2  4-  3^c2  4-  ^c2d  4-  6^&r/  4-  d3  4-  3«r/2 
4-  3W2  —  3-Y/3  —  6«#c. 

19.  8^'3  —  36x'2y  4-  i2.«3z  —  2jy3  4-  54a;?/3  4-  2yy2z  4-  z3 
+  6xz2  —  gyz2  —  3^xyz. 

20.  a3x3  4-  yi2bx2y  4-  3a3.r3.i'  4-  3a*  mx*  —  ia2nx2  4-  6bmyz 

—  6/w»2  —  Gbnnry  —  6bnyz  4-  #3^3  4-  ytb2xy2  4-  $b2y2z  4-  $b2my2 

—  3^2»//2  4-  damxz  —  6amnx  —  Gnnxz  4-  zz  4-  3«#23  +  3^'2 
+  3^?22  —  3^23  +  dabmxy  —  6abnxy  4-  w3  +  ynn2x  4-  2>bm2y 
4-  3?«2«  —  3m2w  4-  dabxyz  —  n3 4-  ^an2x 4-  3^3<y  +  3 ^22  4-  ynn2. 


1.  0s  4-  #*. 

1         1 

2.  rr-  —  2;^. 

3.  an  zt  xn. 

4.  a2  4-  2«j'  4-  .t3. 

5.  a3  +  sa2x  4-  3«^'3  +  *3  = 

sq.  root  ; 
a2  4-  20X  4-  x2  —  cube  root. 


rage  109,  Art.  278. 
6.  a2  +  x2. 


7.  a*  4-  i- 

8.  a  —  x. 

9.  rr  —  1. 
11.  3  —  -\Aj. 
1?.  6  +  V5. 
*3-  5  ±  ?  Vs 


)U 


Fafjf'S  110-122, 

2.  3  (i  +  i  V=~3)> 
3-    a(*±  W-3)- 

4.—5  and  5  (£  ±  J  V—  3) 


ANSWERS. 

.-lrf*.  305-310. 

6.  a. 

7.  5  Vo.r  —  2  \/—  «&• 


9- 
10. 


II. 

12. 

i3- 
14. 

i5- 
16. 

i7- 
18. 
19. 
20. 
21. 
22. 
23- 
24. 


JPaflre 

22?. 

a3b  +  «&3  —  «2£  +  ab% 
4-  rf&fti  +  tffai. 

a*  4-  fo 

t  z.1 

a-  -t-  o-\ 
a  —  Z>. 
a  -  b. 

(a+»)(tf*+8*)(«t*+«*). 

■V —  «&cd. 
8a3. 

a2  +  ». 
a  +  V —  x. 
V —  x  —  V —  a. 
a  (a*  —  Xs). 
a*  +  **. 

m* +  «=   *w5  +  w2         1  +  re   l+»i 

a~mn~b  mn    +  a  m  b  n 

l  +  m  1  +  n  m  +  n   m  +  n 

-f  a~n  b^1    +  a  mn  b  mn  . 


8.  2  V«2  —  tf2. 

25.  an  —  A". 

26.  (P  +  aJs  4-  Ja. 

27.  a3  —  ofib  4-  62. 

28.  cfi  +  a25*  4-  ff^Jt  4-  ab 

1,4  ,5 

+  a*b*  4-  A  :. 

X'l  12,2  8,4 

29.     fl   »      -    ft  5  0>    +    rt^5 

*  t  ■  -78 

—  tf*ft*  4-  //\ 

2-2  20  7  1  -  ,  1 

30.  « »   —  a"3  0*  +  a6P 

16,8  1_4,  .,5 

—  cf*0*  4-  aa 0  —  afyi 
4-  cf^ft*  —  a^fa  4-  a2^2 

—  a*o*  +  a36-  —  err . 

31.  1024. 

32.  1024. 

33-    i°24- 
34.    1024. 


2.    2  V3. 
3    ii- 


3-  3  /V/2- 

4-  1^3- 


Prtflre  125,  4r*.  3i2. 

4.  —  12. 

5.  —  6  y/2. 

Art.  313. 

5.  vV 

6.  1  4-  -v/^a. 


ANSWERS. 


325 


i.  a. 

2.     I. 


Page  12S,  Art.  31S. 


4CIX4 


2  a/ i  +  x 
3- " 2- 


7.  -V2+|  V—  I. 

8.  Zero. 

2  a2 
9- 


a2  —  a;2 
5-  J3- 
6.  2  +  3  V3  —  a/5- 


,P«</e  129. 

14.   — 


10 


62 

205 
X 

11.  Zero. 

12.  —  iV—  3° 

13.  —  2. 


«# 


./ 


8a;  V1  —  # 
15.  Zero. 

6    V(i  +  a*)*  +  x  = 
2  (1  +  a?)* 


1  + 


a2—  a9 

19.  a  —  x. 

20.  a  +  22!  —  b. 

21.  tf6c2  V—  «• 

22.  I. 

23.  oi*c*. 

24.  aP»+«J»+»v+*. 


17.  («ic+a2a;a+«— »)  ^/a—x. 


Page  130. 

(m  +  ny   (m+n)3  m(n*  +  \) 
2K.    (I     mn    b    r",n    C       n 

26.    fl^  +  =-• 


!7.     — - -• 

/         a  —  b 

_    ^  —  ot/3  +  ws 
,8.   -* sL. 

a;3  4-  y 


29. 

3°- 
3i- 
32- 


6»3  (a*  +  tf*ft*  +  aJ  +  d*bi  +  «3 §3  4.  j¥j 

A3  -  a2 

,3  1  11  3\        o 

(aj*  —  £?/*  +  a;-y2  —  y*)  x2 

x%  —  y 
a:2  (x*  +  gyt  +  gffi  4-  yf ) 

a;2  —  y 

87  S_4     1  81  1_8     3,  „    „  1 2     R       ,  9     „ 

x~~»~  4-  a;  sy^  4-  x  s~y  4-  a;^"y^  4-  x*y2  4-  af*  «/*  4-  %sy 


-f 


''  '  S         A  S 

x  ■'//-  +  .*'57/4  4-  ?/- 


x6  —  y5 


326 


A  X  S  \V  B  B  S . 


33- 

34- 
35- 


2_8  2_6      2  0      4  2  2      fi  2_p      8  .     n  \_8      1_2  1_4      1  4 

x* — x  3 'yv+xSys— xs  ~y*  -\-x*  //••  —  r'y'  +  x  ■>  y  *  —x  *~y* 


+ 


,      IS  10      11     ,  8       .  „      2  2      ,  4      2  4  2      2  6  8  8 

°  y "'  — x a y 5  4-a^y4 — ^// :'  +-':i// 5 — ■i"jyz  +y s 

z10  +  #6 


1      o  A  34  „  £     £.  „      1 

a^  —  ay3  4-  a^t/a  _  azy  +  a"-?/3  —  a3//j 
y*  —  a3 


1'age  131,  Art.  319. 


i.  Given. 
2.   x  —    I. 


3-  a  =  25- 

(a  -  6)* 


4.  x-  = 


2(7  —  b 


1.  .?-  —  5-T  —  14  =  o. 

2.  2%  —  (a  4-  £)  X  4-  tf£   =  o. 

3.  x%  —  (a  —  b)  x  —  ab  =  o. 


/    ab    Y 

6.  x  =  3. 

7.  X  =■  VIC. 

8.  x  =  cc,  or  ±  5. 

n       r    2  5 

9.  X   TS. 


Page  135,  Art.  330. 

4.  x2  4-  (a  +  b)x  +  ab  =  o. 

5.  .<-  —  6x  +  12  =  o, 

6.  2*  +  4>r  +  5  =  o. 


1.  *  =  ±  f. 

2.  x  —  ±  3. 

3.  aJ=$[a-J±(«  +  &)V^]. 

4.  x  =  3,  or  —  f.^ 

5.  ./•  =  2  ±  V—  23. 

6.  .r  =  5,  or  f. 

7.  a;  =  ±|  V2. 

8.  'x  '*=  I  (a  4-  b  4-  c  ± 

y (fi  +  p  +  (.-i  _  r/£  _  ac—ic). 

9.  .r  =  ±  5. 

10.  x  =  1,  or  —  2. 

1 1.  .?  =  ±  V—  be  =  —?Vbc, 
or  —*Vbc. 


rage  137,  Art.  335. 

12.  .r  =  *  (1  +  a/i7)- 


13- 

x  =  \(a  ±  v«2.  +  b). 

14. 

x  =  3,  or  —  4. 

is- 

x  =  \,  or  —  f. 

16. 

x  =  1  (9  ±  V^45)- 

i7- 

*  ±a 

18. 

;z2  —  mr 

±  Va?  {m*  —  n*)  4-  As>n2]. 
,9.  s=  V»(i±^)- 


ANSWERS. 


327 


i.  15  and  6, 

or  35  and  —  14. 

2.  10  and  4. 

3.  17  and  9. 


Par/c  138. 
PROBLEMS. 

7.  36  and  64. 


4.  \ 


'a2  ±  b2 


5-  36- 

6.  58  and  37. 


4  miles  per  hour. 
10. 

2  2 

25  miles  from  0. 
n. 


13.  5000  in  1840, 
4000  in  1850, 
5200  in  i860, 
5300  in  1870. 

14.  12  and  18. 

15.  144. 

16.  3  inches. 

17.  The  man  36  yrs.,  son  16. 


Page  139 

18. 


[n— a±V{ct— n)2  —  4b]. 

)       r 

hV3\and 


m 

2 


's/n  —  1 

y/411  —  1  —  V3 

Vw —  I 

19.  7. 

20.  75  at  $4.  or— 120  at  —  $2.50. 

21.  256  sq.  yds. 


JY/rye  i^/,  Art.  339. 

1. » =  a  ±  \  Vi  —  4")w- 

2.  a?=  (-liivr+^r. 

3.  «  =  ±  ^«2  -  [-  \  ±  JVi  +  8  {m  4-  wTJ2. 

2  0  4 

5.  x  =  27,  or  —  64. 

6.  a;  =  64,  or  729. 

•     \  ±  Vn(n  —  1)  +  i 

7.  x  —  :^ -—       — » 

I  ±  Vn(n—  1)  +  i 


Prtgre  i^2. 


8.  a:  =  4,  or  —  1, 


9.  .?  =  ± 


20 

V3 


328 


8 

ANSWERS. 

IO. 

x  =  o,  or  —  \  (b  ±  V  b2  —  A°)- 

1 1. 

x  =  5,  or  26|. 

12. 

x=  ±i  V3. 

13- 

a;  =  3,  or  —  £. 

14.  a;  =  —  (1  ±  2  V—  2)6,  or  o. 
729 


15.  X  = 

0,   i  or  2. 

16.  X  = 

±  1. 

17.  X  = 

18.  a;  = 

—  1,  or  \  (  — 

I  ± 
I  ± 

V- 

3),  or|(i±  V—  3) 

19.  x  = 

±  3«- 

20.  a:  = 

2. 

21.  a;  = 

22.  £   = 

a  (1  ±  ??)2 

1  ±  2W 

2  aft 

23.  S  = 

±i- 

Pr/flre  i4S,  Art.  350. 


5.  a- 

6.  x 

7.  a; 

8.  3; 

9.  x 

10.  .r 

11.  x 
V 

12.  .T 

13.  X 

14.  # 

15.  a; 


5;    V  =  i- 

7,  or  2  ;     y  =  2,  or  7, 


y  =  S 


;.T /*==). 


3,  or  2  ;     y  =  2,,  or  3. 

±  2,  or  =F  ^6-  V5  ;    y  =  ±  3,  or  ±  f  V5. 
2,  or  i  ;     7/  =  3<  or  —  24. 

4,  16,  or  14  ±  V58  ; 

5>  —  7,  or  —  1  ±  V^8. 

\  (1  ±  V—  7) ;   y  =  i(iT  a/1^). 

4,  or  —  2  ;     ?/  =  2,  or  —  4. 

±3;   y  =  ±i- 

±2;    y  =  ±i. 


ANSWERS 


329 


1 6.  x 

y 

17.  X 

18.  X 

19.  X 

20.  X 

21.  .r 

22.  a; 

23.  a; 

24.  x 

25.  iC 

26.  a; 

27.  z 

28.  iB 

29.  a; 

30.  a; 


I  [a  ±  Vcfi  +  b  ±  *Jb  -  2  (a*  ±  a  V«2  +  &)]; 
i  [a  ±  Va*  +  b  T^i-2  («2  ±  a  y^"+T)]. 


1,  or  —  2  ;     y  =r  —  2,  or  1. 
16,  or  4  ;     y  =  4,  or  16. 

9,  or  1  ;     y  =  1,  or  9. 
2  ;     y  =  1. 

4;     2/  =  2. 

2,  or  4  ;     y  =  4,  or  2. 

5  ;    2/  =  3- 

±  6,  or  +  3  ;     y  =  ±  3,  or  ±  6. 

±  5 ;   y  =  ±  !• 

2- ;    y  =  1. 

4,  or  32  ;     y  =  —  3,  or  f 

3,  2,  or  —  z  ±  -j  Vio  ; 
2,  3,  or  —  2  +  i -v/xo. 

5,  or  I ;     y  =  2,  or  ± 
2  5    #  =  3- 


2.  7,  it,  and  23. 

3.  6,   13,  and  25. 


Page  149. 

PROBLEMS. 

4.  4  and  5  yards 
5-  36. 


Page  150. 

6.  15,  12,  and  9. 

7.  A's,  $80  at  5%;     B's,  $120  at  6%. 

8.  A's,  $100  ;     B's,  $150. 

9.  ,-u =  ( —  an*  +  b  \/b2m2  —  a2)n2  +  n*),  and 

(r  —  a4 


Ifi  —  a* 


-s  (—  £«2  ±  a  Vb*m*  —  a2m?  +  w4). 


330  ANSWERS. 


10.  The  parts  of  a  are 


—  \ab  —  m  +  n  ±  \/(ab  —  m—  n)2  —  4m >i\,  and 
20 

—.  [ab  +  m  —  n  ^f  V{ab  —  m  —  nf  —  4m a]. 
20 

The  parts  of  b  are 


211 


[ab  +  m  —  n  ±  V(ub  —  m  —  nf  —  41/in],  and 


—  [ab  —  m  +  n  ^  V(ab  —  m  —  n)2  —  4inn~\. 
211  L 

11.  yTi. 

12.  i  (3  +  V^~3)  and  ±  (3  -  V^3). 
13-  i(3  ±  V5)  and  i(i  ±  Vs). 

6  ±2  V5  6  ±  2  V5 

'5-  ±  3  and  ±  1. 
16.  A  had  200  acres,  at  $1.50  per  acre  ; 

B  had  400  acres,  at  $.75  per  acre. 

Page  153,  Art.  357. 

1.  2.  Given. 

3.  x  >  4,  the  limit  of  #. 

4.  2;  >  2£,  the  limit  of  a;. 

5.  x  >  «,     a:  <  #,  the  limits  of  x. 

6.  x  <  5,     a;  >  3  ;     .*.     4  is  the  only  integral  value  of  x. 

7.  x  <  6  and  »  >  4 ;     .•.     3,  4,  and  5,  are  the  integral 
values  of  x. 

8.  The  number  is  19  or  20. 

9.  He  sold  60  apples. 
10.  20  and  5. 

Page  100. 

P  R  0  B  L  E  M  S . 
i.  The  third  proportional  is  100. 

2.  The  first  term  is  54. 

3.  The  mean  proportional  is  15. 


ANSWERS.  331 

T     abc   , 

4.  In  -7T-,  clays. 
*         ab       J 

5.  16  and  20  tons. 

6.  A,  54!  hours  at  9^  miles  per  hour  =  494-ff  miles  ; 
B,  59!  hours  at  nT\  miles  per  hour  =  57oT6¥  miles  ; 

A,  3  hours  at  o  miles  per  hour  =  o  miles  ; 

B,  o  hours  at  2  miles  per  hour  =  o  miles. 

7.  The  numbers  are  30,  48,  50. 

8.  6,  or  3  (—  1  +  V—  3) ;  and  9,  or  §  (—  1  ±  V^~i). 

9.  The  number  is  863. 

10.  The  numbers  are  a,  2a,  and  3a. 

11.  The  conditions  are  not  independent. 

Page  162,  Art.  381. 

1.  Given. 

in 

2.  x  =  -2 ;     xif  =  m  ;     »r  :  x2  =  y22  :  3^2. 

3.  a;  =  m  (a  +  y);    xI  :  :r2  =  a  +  yT  :  a  +  y2- 

4.  x  =  m  (tf  +  if)  ;     xx  :  x2  =  yY2  +  yz3  :  y22  +  y23. 

5- «  =  rr+ya ;  a?i :  x*  =  y*  +  y*  :  2/1  +  vi  • 

6.  The  values  of  y  are  25,   nf  ;  of  x,   12,  4f. 

7.  The  values  of  x  are  ff ,    2?>  xs>  °- 

8.  The  value  of  m  is  1. 

9.  The  value  of  y  is  f  \  2X%. 

10.  y  =  2(s  +  i). 

11.  19200  inches.    5  seconds. 

Paflre  iOo,  ^?*f.  35*. 

1.  362880.  3.   126. 

2.  15120.  4.  126. 


332  ANSWERS. 

_  n  (n  —  i) (n  —  m  +  i) 

5*   °»»  — 

n  \ll_ 

n  (n  —  i)  . . . .  (n  —  m  +  i)  \u  —  tit 


^n — m  — 


\n  —  m  \m 

n  (n  —  i)  .  .  .  .  [n  —  (n  —  m)  +  i] 

to  —  m 


\n  —  m  \m  ' 


n  (n  —  i) . . .  (m  +  i)  \m 


ii 


\n  —  m  \m 


6.  a  =  i5. 

7.  Cs  =   84. 

8.  No  more. 

9.  [m. 

10.  Zero. 


\n  —  m  \m 

11.  8,  or  —  1.     The  2d  Ans. 

not  applicable. 

12.  m  =  n. 

13.  m  =  n,  or  w  +  1. 

lS-  4540536°o- 
16.  3991680. 


Page  175,  Art.  417. 

-6.  Given. 

-£  =  (x-b)  (x-c)  +  (x-a)  {x-c)  +  {x-a)  (x-b). 

(It/  X2   +    2X  —    I 

dx 


(X  +   l)2 


9-  iz  = 


1  —  xA 


dy 

dx        (x2  +  i)2 


10. 


dy 

dx 


to"  - 


2  >    r/2//  _      3 


a;^       ^ 


t/a;2 


+  -!  +  —• 

4x?       x3       535 


(///  1 

11.  — —  =  • 

dx       (1  —  a;)2 

12.  ^  =  5  +  2Cx  +  zDx\ 


i3- 
14. 


dx 


r?>/ 


t/.c 


;  =  —  4  (1  —  z)3- 


ANSWERS.  333 


J    da; 

16.  ^  =  -  TO  (i  -  xT~\ 
dx 

17.  -^  =  7a;6  —  20a;3  +  6a;. 
dx 

18.  g  =  (*-i)  {x+2)  (x-5)  +  (*-i)  (.,+  2)  (*+3) 

+  (*-i)  (*-S)  (*+3)  +  (3+2)  (a-5)  (*+3). 

19.  Given. 

rfy        \/«  (y  —  %) 


20. 


«#        2  V#  Vxy3 


dx  _  1 

"'  ^  ~"    ~  2  (1  +~^i* 

22.    4  = £— . 

f&e        (1  —  a;2)-3 
efy  2a: 

23'  ^'  "~  3  («  +  «2)f' 

24.  du  =  2xyhlx  4-  yxh^dy  +  $x2y2dx  4-  2X3ydy. 

du 

dx 


25-  3=  =  *3  -  » 


2.  Given. 

3.  a;  —  a;2  4-  a;3  —  as*  +  etc. 

4. 1  +  x  —  a;3  +  etc. 

x 

5.  a? 1 - —=  -  etc. 

2a;*       2  •  4a;3        2  •  8a;2 

.    a        a  —  1        .  ,  . 

6.  —  H (-  (a  —  1)  (1  +  x  +  aH  4-  etc.) 

xl  x  v 

1  a;  xx% 

7.-7 i  +  — - 7  -  etc 

a3       2«s       2  •  4«s 

8.  l—-x  +  ^x*—  1  a;3  4-  etc. 
a       a^  a3  a4 

9.  —  1  —  2a;  —  2a-2  —  a;3  4-  2a;4  4-  7.T5  4-  etc. 


334 


ANSWERS 


IO. 
1 1. 


13- 

14. 

IS- 

16. 

17- 
18. 
19. 

20. 


1  +  x  +  x~  +  .r5  +  jc*  +  et^. 

1  —  3*  +  3^  —  33*  +  3-i;5  —  etc. 


«  ~~  a2  +  a3 "~  ^  +  etc* 


X        X2        xz 
,  a       a1       a3 

i         I           l 
* i i  - 


—  etc. 


Sizs 


3*1 

1 

1  +  z*  +  x  +  »*  —  xi  +  a4  —  etc. 
—  a*  +  32*  —  5.r  +  73$  —  92$  +  etc. 
1  +  x*  +  a$  4-  a*  +  x  4-  ajf  +  etc. 
#3  -j-  #*  +  s£*  +  cc»  +  etc. 


1,  2.   Given. 

3      3 

4 


+ 


Page  183,  Art.  430. 

3i  1 


r4  (*  —  2)       35  (•'•  +  5)        10* 
J_  5  1 

3  (X  +    1)2  +  30JT+1)  ~~  X 

£ . X  +  2 

2  (X  —  i)  2  (z2  +  3  +  4)' 


+ 


672   —    I05.T  48.T  —    TOO 


49./-       49  (a-2  +  2x  +  7)2       49  (/2  +  2X  +  7) 

"3  J  25 


40  [x  —  5)       sx       4o  (»  +  3) 


4-  1. 


+ 


a  —  a*        a4 
x  —  1    '    (x  —  i)2       x 
1  1 


4  (3  —  1)        4(*  +  l)        2  (a2  4-  1) 
122 


f 


+ 


9  (X  —  2)2  27   (.?  —  2)  27   (.1-  +l)  9  fa  +    l) 


ANSWERS, 


335 


II. 


(a  +  b)  {x  —  b)       (a  +  b)  (x  +-  a ) 


+ 


7 


r.  + 


23 


9  (a  +  2)       x  —  2       4  (a;  —  2)2       12  (a;  +  2)2 
5  3 


+ 


18  (a;  —  1)       2  (a-  4-  1) 


Page  ISO,  Art,  433. 

a5  -f  5«4&  4-  io«362  4-  iort2&3  4-  5«J4  +  b5. 


2.  «■ 


1 
2  a3 


i2. 

8ai 


J" 


i6«'3 


-.  —  etc. 


1        35       6b2       to63       1564 


i_5       & 

„2  "T"   „3 


rt 


2/ 


£3 
r4  +  etc. 


a 


1 

1  + 


8a;3 


5?/3 
i6a;; 


+  etc. 


^+2I  +  f2+f3+etc. 

+ 


x 


X*      ' 


4n6 


%        zn 

m3  4 :  — 

3?«3       gnfl       iSivi* 

I  271  5W2  40»? 

s«  - i  -1 5  H 1  + 

ma       3m3       9/n3 

1  2)i  5  M2 

w*      3m3       9m3 

I         4  4  32 

fo.  za  -  -2j  -  — 4  -  — * 
3a;s       93;?       81a;3 

a3  +  6a-2  4-  12a;  +  8. 


7W4 

n>  +  etc. 

243m3" 


ti  +  etc. 
tirm  3 

4cm3 

-^— -  +  etc. 
81m"3" 


—  etc. 


11. 

12. 

i3- 
14. 

IS- 


11         1  1          q 

**  + j  4-  —  -  -2,  +  etc. 

a3       2a;3  2a:2       8a;* 

«5a;5  4-  5  «4ia4?/  +  1  oazb\r%if  +  1  oa2b3x2ys  4-  5  a£4a;?/4  4-  J5//5. 

11           J?/  S2?/2             Z»3?/3 

t  r  5  3   1         5  E       etc. 

2«3a;^  8a3a;3"       i6f/3a-3 


arx*  4 


11  S     8    "I       „     5      6  ,      7      7     ~t~     elt" 

^a;3" 


8      8       '        _      5      6  -      1      7 

2^-a;3       8a*a;*       ioflW5 


3:36 


A  N  S  W  EKS. 


16.  25a2  —  loax  +  49.T- 


1    1  2X 

17.  3  «    +  —T 
3-«J 


2.T2  4^ 

-7-8   +  ~ 7  -   et°- 


18. 


19. 


5  .  .  .  .  (2??  —  s)    ,        ,     7     * 


2«— 1  .  1   .  2   •  3  ■(?£  —    i) 

■J -A-       S2n    ~  3)fli-na*-i. 


>«— 1  •  I   •  2  •  3  -(M  —  i) 


Plriflre  10?,  4r«.  11<S. 


1-3.  Given. 

4.  0. 

5-   *• 

6.   -5 

7-  —  3- 

8.   —2. 

9.   —  1. 

10.  0. 

1.  2. 

12.  4. 

13.  6. 

2.  1548.781. 

3  1-973355- 

4.  —  in. 

6.  78. 

7-  -°375+- 


Prtf/e  106%  4« -t.  459. 


8.  14.385416  +  . 

9.  2.7234  +  . 

10.  2906.25. 

11.  .1S13+. 

12.  —  4.619-f . 


Page  197,  Art.  401. 


13- 

Given. 

14- 

.00033215 

J5- 

33*333"  + 

16. 

191. 8139. 

17- 

i-3- 

18. 

5.23176  +  . 

19.  I.0S37+. 

20.  2.5047+. 
2  1.  2. 1248-)-  . 

23.  0.342S4  +  . 

24.  0.54613  +  . 

25.  0.32471+. 


Page  215,  Art.  497. 

n         yi~  —  n 
i.  an  =  yi  —  2  ;     S„  - 

2-  Oh  =  3l 


2   }  '    »     

4 


AN  S  WEES.  337 

3-  On  =  5n  —  3  ;     »»  = 

3?i2  _  7?i  _|_  6       c,        w  (ft2  —  211  +  3) 

4.  an  = ;     &n  — 

11  {n  +  1) 

5.  an  =  n;    Sn  =  — ^ 

6.  On  =  2W  —  1 ;     Sn  —  n2. 

7.  an  =  2)i ;     tf»  =  n  +  w2. 

ra(rc+i)        ~    _  n  (n  +  1)  (w  +  2) 

8.  flfl  _  -         ;     ^  _         -  j  .  2  .  3 

W(W+I)    (2»+l) 

9.  «w  =  1l2;      bn  =  - 


IO.  tfra  =  «3 ;      £w  = 


1-2-3 

112  (n  +  i)2 


w(w+i)(w  +  2).     »     _  »(w+i)  (w  +  2)(»  +  3) 

11.  r^  — j     Ob  —  :  ' 

1-2-3  1-2-3-4 

n(n  +  1)  (11  +  2)  (n  +  3) 

12.  «,t  =  — — 

1-2-3-4 

n  (n  4-  1)  (w  +  2)  (w  4-  3)  (n  +  4) 
^   __ _ . 

1 .2-3-4-5 
11  (11  +  1)  (n  4-  2)  (»  4-  3)  (w  4-  4) 
13.*= x-2- 3-4-5    -      —  5 

w  (w  +  1)  (11  +  2)  (;?  +  3)  (n  4-  4)  (^  +  5) 

14.  (1)  o.         (2)  o.         (3)  o. 

15.  (1)  The  eighth.        (2)  The  ninth. 


Page  216. 

16.  220  balls  in  a  triangular  pyramid. 
I7-  385  balls  in  a  quadrangular  pyramid. 

18.  280  balls  in  an  oblong  rectangular  pile. 

19.  The  —  10th  term  is  210. 

20.  (1)  304  ft.       (2)  1600  ft.        (3)  900  ft.        (4)  ^^  ft. 
(5)   i6n2  ft. 

15 


338  ANSWERS. 

Page  223,  Art.  513. 

i.   n20  =  524288,  S20  =  1048575; 

an     =  2"-i;  Sn    =  2»  —  1. 

2.  aI0  =  59049,  tfjo  =  88572; 

««    =  3n;  sn    =  f  (3»  -  I). 

3.  «10   =  2560,  £I0  =  5115  ; 

On      =S-2n~1;  Sn      =S(2»  — I). 


Pa</e  224. 

4. 

#00    =    O, 

&  =  16. 

5- 

«»  =  °> 

fl.  =  5i 

6. 

m'  =  1, 

^5?   =  TTS' 

7- 

W'   =    2, 

a,j  =  2,           «lS  =  4, 

a2l  =  16, 

ff2l     =     32- 

S. 

"10  =  — 

19683,          #I0  =  —  14762; 

as     =  81 

Ss    =  61. 

9.  The  scale  is  22s,  —  3a;2,  3a;. 

Series  continued  is  86a;7,   171.T8,  341a;9,  672a;10,  etc. 
10.  The  scale  is  —a;2,  2a;.        «I2  =  35a;". 

Page  227,  Art.  520. 

1.  The  fifth  term  is  ^. 
„    1     i      t      1 

2-  3>    To?    T*>    TF- 

3-  5.  6|,  8|,   i2i    25. 
4.  See  Key. 

Page  227,  Art.  521. 

1.  a  =  p  (1  +  r)*. 
rs  ( 1  +  r)* 


2.  p  = 
3-  *  = 


(1  +  r)<  -  1 


(1  +  r)' 


ANSWE  RS. 

Page  228. 


339 


4- 

5- 

6. 

7- 
8. 

9- 

io. 


12. 

13- 

14. 

15- 

16. 

17- 


w  =  7- 


(1  +  r)< 
a 
"r(i  +  ?•)' 
g[(i  +  r)y  — 1] 

w  -       r  (1  +  r)^' 

$595.58. 

$871.73. 

£783.53- 

_  a[(i  +  r)m—  1] 
w  ~"        r(i+r)u 

a  [(1  +  r)7  —  1] 
W~        r(i+r)« 
$1245.43. 
$1666.66. 
$2230.38. 
April  27th,   1875. 

Answers   in   order  in   miles    160,  55,    55,   10,  —  n, 
—  270,   —  4. 
40.951  miles;  61.25+  miles;  2.73+  days;  1 00  miles. 


18.  00  ,  300  ft. 


5 

"2T> 


1 

T2- 


Page  229. 


Page  230,  Ari.  522. 

7-    ^. 

8.    1. 
9-  tV 


10. 
11. 


7 


?0- 


-P«</e  2Si,  Art.  523. 

x  =  y  —  2if  +  5y5  —  91/  +  etc. 

x  =  1  —  y  +  (1  —  ?/)2  +  (1  —  ?/)3  +  (1  —  yY  +  etc. 

*  =  y  +hf  +  if  +  2\f  +  Thy5  +  etc. 

x  =  y—i—2(y—iy+s{y—i)s—9(y—iy+  etc. 


340 


A  S  S  W  E  R  S  . 


Page  243,  Art.  542. 

2.     Z~  —   1$Z5  +   848Z2  —  2432  —   729   =   O. 

3-   z4  —  3*3  +  45*2  +  125  =  o. 
4.  z4  —  $z3  —  7Z2  +  8  =  o. 

Par/e  244,  ^4»£.  545. 

1.  —  1,     4,     and     5. 

2.  1,     —  1,     2,     3,     6. 

3-    i- 

4.   2,     —  2. 

5-  5- 
6.  None. 

Prtgrc  246,  Art.  548. 

1.  re6  —  2X5—  37a4  +  68a;3  -f  327a-2  —  450a;  —  675  =  o. 


2.  a4  —  a 

x3  +  «Z> 

a:2  —  «Jc 

x  +  abed  = 

0. 

—  b 

+  ac 

—  aid 

—  c 

+  ad 

—  acd 

-d 

+  £c 
+  hd 
4-  ctf 

—  hed 

3.  x3  —  8  =  0. 

4.  xi  +  6.r3  +  7a2  - 

-  24a;  —  44  =  0. 

5.    .'E6  —  Sx 

5  +  31a4 

—  402s  + 

6xz  +  288  = 

0. 

Page  251,  Art.  553. 

1.  4  positive,  1  negative,  2  imaginary. 

2.  2  positive,  2  negative,  2  imaginary. 

3.  o  positive,  1  negative,  4  imaginary. 

4.  1  positive,  o  negative,  4  imaginary. 

raye  262,  Art.  567. 

1-6.  Not  desirable  to  give. 

7.  z4  —  6z3  +  28Z2  +  2  —  16  =  o.         z  - 


2X'%. 


ANSWERS.  341 

8.  z%  —  6z3  +  28z2  —  482  —  32=0.        z  =  2xK 

9.  zn  —  642s  —  3842*  —  5122s  —  4096  =  o.         z  =  22T&. 

10.  25  +   22T4  +   243^  —   I3122    =0.  2   =    9^8. 

11.  Z2  -\-  Z  —  6=0.  Z  =   2X. 

12.  X  -J"   I    =   O. 

13.  All  the  roots  are  included  in  the  equal  roots,  and  the 
result  of  removing  them  is  1  =  1,  an  equation  of  the  zero 
degree  and  having  no  roots. 

14.  The  same  as  the  last. 
The  equations  are  as  follows  : 

1.  x3  —  2X%  —  5a;  +  6  =  o. 

2.  x4  —  7a:3  4-  10a;2  4-  142'  —  24  =  o. 

3.  x3  4-  5a;2  4-  2a;  +  10  =  o. 

4.  2X*  —  a;3  —  i6x  4-8  =  0. 

5.  x*  —  8a;3  4-  27a;2  —  46a;  4-  44  =  o. 

6.  a;6  —  12a:5  4-  58a;4  —  144a;3  4-  193a'2  —  132a;  4-  36  =  o. 

1      -r4   _l_    o  ?  ^3  _1-    I  1 0  7.2  4  0  9,  r     1        35      —    ,-, 

7.  x    •+-  2-g-a    4-  -35 -x   —  Tttx  -r  T4  4"  —  °- 

8.  6x5  —  41^  +  97a-3  —  97a;2  4-  41a-  —  6  =  0. 

9.  a4  —  4  =  o. 

10.  x5  4-  ioa^  4-  40a;3  4-  80a;2  4-  80a;  4-32  =0. 

Page  266,  Art.  575. 

2.  x  =  —  3.84-. 

3.  x  =  6.5+. 

4.  x  =  0.3  4-. 

5.  x=—  i.o+,     1.3  +  ,     and    4.6+. 

6.  a?  =  —  1.3  +  . 

7.  a;  =  ±1.4+     and     ±  J-7+- 

8.  x  =  1.4  +  ,     5.24-,     and  -  —  0.64-. 

9.  x  =  —  6.64-. 

10.  x  =  0.4 4-,    0.74-,     and     —6.2+. 


342  ANSWERS. 

rage  271,  Arts.  578,  579. 

2.  X  —    I.3797+. 

3.  x  =  125. 

4.  x  =  0.601+     and     —  1.6  -4-. 

5.  x  =  8.8  +  . 

6.  x  =  1. 259921  +. 

7.  x  =  0.90 +  . 

8.  x  =  1.3  +  ,     1.6  +  ,     and     —  3.04  +  . 

9.  #  =  2.6  +  . 
10.  a;  =  — 2.5+. 

Page  277,  Art.  589. 

1.  Given. 

2.  sb  =  i,     or    \  (1  ±  V—  3)- 


3.  a;  =  ±  (3  ±  V-  7  ±  V-  62  +  6  V-  7). 

4.  x  =  ±  1,     or    i  (—  7  ±  3  Vs)- 

5.  a;  =  1,    or    ^  (—  1  ±  V'71  ±  v—  72  +  V'71). 

6.  x  =  ±  1,     or    i  (3  ±  V^5  ±  ^8  +  V^f). 

7.  <b  =  $  (1  ±  V^3  ±  V-  !8  +  2  V^s). 

8.  a=  ±  1,    or    i(i  ±  V^l). 

9.  #  =  +1,    or    -  (1  +  -\A  —  a2). 

10.  a  =  —  TTo  (—  4  ±  V—  29  +■  V_  113  +"  8  V-T9). 

-Pa<7e  27S,  -4r*.  592. 

1.  Given. 

2.  a?  =  ±  1,     —  £  ±  J-  V^l,     and    \  ±  *  V^, 

3.  x  =  ±  1,     or     +  V—  t,  for  n4  —  1  =  o ; 

x  —  ±  V  ±  V—  1,  for  'x4  +  1  =  o. 


ANSWERS. 


343 


4.  x  =  ±  1,     or     ±  i  (1  ±  V—  3),  for  x«  —  1  =  o ; 

a?  =  ±  V^,     or     ±  V|(i -j-  V^3),  for  ar<5+ 1  =  o. 

5.  See  Example  1  for  a;5  —  1  =0. 

aj=— 1,  or  i{i±V5±^—  io±2a/s),  for  z5+i=o. 

6.  Multiply  the  roots  of  a;5  +  1  =  o  by  jk 

7.  Multiply  the  roots  of  a;5  —  i  =0  by  4^. 

8.  Multiply  the  roots  of  xG  +  1  =0  by  5*. 

9.  Multiply  the  roots  of  xfi  —  1  =  o  by  3*. 

Page  279,  Art.  593. 

1-2.  Given. 

3.  x—  1. 7558749  +  . 

4.  x  =  1.3  +  . 

5.  £  =    1.9  +  . 

6.  »  =  1.6 +  . 

7.  In  12+  years. 

.    T     log  A  —  log  a 

8.  In  -^ — T 5^ h  1  years. 

log  (1  +  r)     ^     J 

9.  In  26.75+  years. 


Pteflre  ««1, 

Art.  596. 

I- 

-2.  Given 

n       5 

3- 

7 

O         125 
°'     TS'Sl 

4- 

I 
770200* 

9.  Yv 

5- 

3 

IO.     ^. 

6. 

.£    and 

h 

Page  284,  Art.  599. 

1- 

3.  Given. 

4- 

»  =  2, 

or 

1  ±  i  V^g. 

5- 

0?  =  I, 

or 

-  i  ±  i  \ 

^i 

344  A  N  S  W  E  B  S . 

6.  x  =  2,     or  —  i  ±  3  V —  i. 

7.  x  =  4.07 +. 

8.  x  =  3.9  +  ,  —  3.8  +  ,     and     -  0.13 +. 

9.  x  =  2.6+,  0.58  +  ,     and     —3.2  +  . 

10.  :/;  =  1.8  +  ,     —  1.5+,     and     —  0.3+. 

11.  x  =  2.1+,     0.74-,     and     — 2.9  +  . 

12.  2;  =  2.1+,     0.5  +,     and     — 0.6  4-. 

MISCELLANEOUS    EXAMPLES. 

«  +■  5 

26a2  +  38^2; 
35  «2  +  66«a;  4-  27a;2 

3«3  +  gaPz  +  6«.t2  4-  2X3 
2  a4  4-  3«3a;  —  «2#2  —  ^ax5  —  xi 

4.  x  =  1. 

5.  In  30  hours. 

6.  5  hours  27  minutes  16.36  seconds. 

7.  22;  4-  $y  4-  2. 

8.  x  +  2. 

9.  «3  +  3a  4-  2. 

10.  a;  =  7V(—  1  ±  V649). 

11.  30  gallons. 


Prtgre  203. 


12.  9.369  miles,  nearly. 

13.  $1000. 


14.  The  majority  804  ;  the  minority  603. 

15.  x  =  b. 


16.  x  =  2a. 


ANSWERS.  345 


r8.  a  =  ±  i  [±  (VrT^2  -  1)  (VT=^  +  i)]i 

i9.  a;  =  -|  ±  i  a/5- 
20.   In  40  minutes  after  they  started. 

21.  Rate,  3,  5,  and  4  miles  an  hour  respectively. 

Page  300. 

c^x2 

22.  1/   =   7X7 ; »• 

J        ¥  (a  +  x) 

23.  162. 


1 


(1  +  Va)". 

24.  -. 

V2 

25.  IOO. 

26.  See  Key. 

27.  Distance  apart  450  miles.    A's  rate  30  miles  and  B's 
rate  25  miles  per  day. 

28.  The  rate  of  increase  was  fourfold. 

d 

29.  The  first  point  of  meeting  was  at  the  distance  — = 

V2 
from  one  starting  point,  and  the  second  the  same  distance 
from  the  other,  d  being  the  distance  between  the  two  points. 
The  ratio  of  their  rates  is  1  4-  V2. 

Page  301. 

30.  He  had   115   hurdles  and  they  must  be  1.7  ft.  apart 
(nearly). 

31.  12  and  8  gallons. 

32.  x  =  1,     —  4,    and    |(i  ±  V41). 

33.  (2a  —  1)  {b2  —  4C2  _  ab)  =  o. 

34.  The  parts  are  30,  36,  and  45. 

35.  36.  See  Key. 


346  ANSWERS. 

a  (i  -  m") 

37-  w„_i  _  ma' 

38.  23.456  +  . 
39-41.  See  Key. 

Bage  302. 

42.  x  =  2,     —  1.84  +  ,     3.13  +  ,     and     —3.28+; 
V  =  3,     3-58+>     —  2.805 +,     and     —3-77+. 

43.  x  =  —  a  ±  V«2  +  £2; 

V2«2  +   2^  +   V«2  +  ^ 

2/  =  *  ± 2 

44.  See  Key. 

45.  $600000. 

46.  In  1845,  Population  4354 ; 

"   1854,  "         5602; 

"   1862,  "        6943; 

"   1880,  "         1 1478. 

I 


47.  -  y^—r,  \V  ±  W\  (62  +  d'2)];  I  being  the  length  and 

b  and  £'  the  width  of  the  ends. 

48.  The  first  is   2x2  +  4X ;  the  second  is  2XZ  +  $x2  -f  o. 

49.  In  41.06+  years. 

50.  The  chance  is  |^. 


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